Bhishma certainly did that. Drona also fought only half-heartedly for Duryodhana. For example, he pretended that he couldn't kill Ghatotkacha thereby forcing the wasting of the Vasavi Shakti. Also I don't know this for a fact but did Krishna directly tell him "we can't win as long as you are fighting on Duryodhana's side - please tell us how you may be removed from the war".
Drona also fought only half-heartedly for Duryodhana. For example, he pretended that he couldn't kill Ghatotkacha thereby forcing the wasting of the Vasavi Shakti.
This claim of yours is totally wrong. Dronachary was the main cause of Death of Abhimanyu which was the most significant win of Kauravas. Also Dronacharya had killed many warriors of Pandavas. Duryodhana also thought that Drona is not fighting for him but he got a proper reply from Ashvatthama mentioned in Mahabharata, Drona Parva, SECTION CLIX.
'Thus addressed by Duryodhana, Drona's son, that warrior difficult of defeat in battle, set his heart upon destroying the foe, like Indra bent upon destroying the Daityas. The mighty-armed Aswatthaman answered thy son, saying, 'It is even so as thou sayest, O descendant of Kuru! The Pandavas are always dear to both myself and my father. So also, are we both dear unto them. Not so, however, in battle. We will, according to the measure of our might, fearlessly contend in battle, reckless of our lives. Myself, Karna, Salya, Kripa, and Hridika's son, could, O best of kings, destroy the Pandava host within the twinkling of an eye. The Pandavas also, O best of the Kurus, could within the twinkling of an eye, destroy the Kaurava host, if, O mighty-armed one, we were not present in battle. We are fighting with the Pandavas to the best of our might, and they also are fighting with us to the best of their might. Energy, encountering energy, is being neutralised, O Bharata! The Pandava army is incapable of being vanquished as long as the sons of Pandu are alive. This that I tell thee is true. The sons of Pandu are endued with great might. They are, again, fighting for their own sake. Why should not they, O Bharata, be able to slay thy troops. Thou, however, O king, art exceedingly covetous. Thou, O Kaurava, art deceitful. Thou art vainglorious and suspicious of everything. For this, thou suspectest even us. I think, O king, thou art wicked, of sinful soul, and an embodiment of sin. Mean and of sinful thoughts, thou doubtest us and others. As regards myself, fighting with resolution for thy sake, I am prepared to lay down my life. I will presently go to battle for thy sake, O chief of the Kurus.
Also, Drona had not suggested Karna to use Vasavi Shakti instead Kauravas suggested as mentioned in Mahabharata, Drona Parva, SECTION CLXXIX.
At that time all the Kauravas, beholding Karna and that terrible illusion (of the Rakshasa) cried out saying, 'O Karna, slay the Rakshasa soon with thy dart. These Kauravas and the Dhartarashtras are on the point of being annihilated. What will Bhima and Arjuna do to us? Slay this wretched Rakshasa at dead of night, who is consuming us all. They that will escape from this dreadful encounter to-day will fight with the Parthas in battle. Therefore, slay this terrible Rakshas now with that dart given thee by Vasava.
Also I don't know this for a fact but did Krishna directly tell him "we can't win as long as you are fighting on Duryodhana's side - please tell us how you may be removed from the war".
This is not right (AFAIK from Mahabharata) as it is not mentioned in Mahabharata anywhere instead Lord Krishna suggested Pandavas to told Drona that Ashvatthama has been slain. So Bhima killed an elephant named Ashvatthama and told to Drona that Ashvatthama has been slain. Then Drona confirmed it from Yudhishthira he also said the same so Drona became distressed and dropped his arms. It is mentiond in Mahabharata, Drona Parva, SECTION CXCI.
Beholding the sons of Kunti afflicted with the shafts of Drona and inspired with fear, Kesava, endued with great intelligence and, devoted to their welfare, addressed Arjuna and said, 'This foremost of all bowmen is incapable of being ever vanquished by force in battle, by the very gods with Vasava at their head. When, however, he lays aside his weapons, he becomes capable of being slain on the field even by human beings. Casting aside virtue, ye sons of Pandu, adopt now some contrivance for gaining the victory, so that Drona of the golden car may not slay us all in battle. Upon the full of (his son) Aswatthaman he will cease to fight, I think. Let sonic man, therefore, tell him that Aswatthaman, hath been slain in battle.'
After Krishna recites the Gita to Arjuna and before war commences, Yudhisthira gets the blessings of victory from Bhishma, Dronacharya, Kripacharya, and Shalya. Yudhisthira received the blessing of victory from Bhishma and goes to Drona.
Sanjaya continued,--"Then, O son of Kuru's race, Yudhishthira, once more saluting him, accepted Bhishma's words with a bend of his head. And that mighty-armed one then proceeded towards the car of the preceptor ( Drona) through the midst of all the soldiers who were eyeing him, accompanied by his brothers. Then saluting Drona and walking round him, the king spoke to that invincible warrior words that were for his own benefit.
Yudhisthira asked Drona how he will vanquish his foes.
"Yudhishthira said, 'I ask thee, O invincible one, how I may fight without incurring sin, and how, with thy permission, O regenerate one, I may vanquish all my foes.
Drona is gratified.
"Drona said, 'If, having resolved to fight, thou hadst not come to me (thus), I would have cursed thee. O king, for thy complete overthrow. I am, however, gratified, O Yudhishthira, and honoured by thee, O sinless one. I permit thee, fight and obtain victory. I will also fulfil thy wish. Say what thou hast to say. Under these circumstances, battle excepted, what dost thou wish? A man is the slave of wealth, but wealth is not one's slave. This is quite true, O king! Bound I have been with (their) wealth by the Kauravas! It is for this that like a eunuch I shall fight for the sake of the Kauravas. It is for this that like a eunuch I am uttering these words--Battle excepted, what dost thou wish? I shall fight for the sake of the Kauravas, but will pray for thy victory.'
Yudhisthira told Drona to fight for Kauravas but pray for his good.
"Yudhishthira said, 'Pray for my victory, O regenerate one, and counsel what is for my good. Fight, however, for the Kauravas. This is the boon solicited by me.'
Yudhisthira asked Drona how to vanquish him.
"Yudhishthira said, 'I ask thee, O foremost of regenerate ones, listen to what I have to say. How shall we in battle vanquish thee that art invincible?'
Drona then replied that as long as he fights, victory will never be his.
"Drona said, 'As long as I will fight, so long victory can never be thine. (Therefore) O king, seek with thy brothers, for my speedy slaughter.'
Yuudhisthira then asked for the exact means of his death.
"Yudhishthira said, 'Alas, for this, O thou of mighty arms, tell (us) the means of thy death. O preceptor, prostrating myself I ask thee this. (My) salutations to thee."
Drona told him how he can die.
"Drona said, 'The foe, O sire, I see not who may slay me while standing in battle I am engaged in fight, with wrath excited, and scattering (my) arrowy showers continually. Except when addrest for death O king, having abandoned my arms and withdrawn (in Yoga meditation) from surrounding sights, none will be able to slay me. This that I tell thee is true. I also tell thee truly that I will cast off my arms in battle, having heard something very disagreeable from some one of credible speech.--'"
Drona told Yudhisthira that if he cast off his weapons if he hears something disagreeable from some one of credible speech.