Adi Shankaracharya is predicted in the Bhavisya Purana. What about the other Puranas?
Do any other Puranas predict his birth and exploits or make mention of him?
Yes, other Puranas do mention about prophecy of Adi Shankara. As per Kamakoti Site, there are references to Adi Shankara being Avatara of Lord Shiva in Kurma Purana, Markandeya Purana, Aditya Purana, Shiva Purana and Linga Purana.
Among them, i found exact verses in Chapter 30 of Kurma Purana.
तदेव साधयेन्नृणां देवतानां च दैवतम्।
करिष्यत्यवताराणि शंकरो नीललोहितः॥ ३३॥
श्रीतस्मार्तप्रतिष्ठार्थ भक्तानां हितकाम्यया।
उपदेक्ष्यंति तज्ज्ञानं शिष्याणां ब्रह्मसंज्ञितम्॥ ३४॥
सर्ववेदान्तसारं हि धर्मान्वेदनिदर्शितान्।
सर्ववर्णान् समुद्दिश्य स्वधर्मा ये निदर्शिताः।। ३५।।
The people should adore lord Mahadeva, who happens to be the god of gods. The brown complexioned Siva, for the establishment of Srauta (Vedic) and Smarta dharmas (belonging to the Smritis), and for the welfare of the devotees shall incarnate on earth. He will enlighten his devotees on the subjects of the gist of Vedantas, the knowledge relating to the Brahman, and all the dharmas enshrined in the Vedas, which have been narrated earlier.
I will update this answer with verses from above mentioned Puranas after finding them.
Yes there are many Puranas that do. The Kurma Puran Purva Bhag Chapter 30 verses 33-34 states clearly that in the Kali-yuga, lord Rudra (known as Nila-Lohita) will incarnate on earth in order to reestablish the principles and doctrines of the Shrutis and Smritis in order to bring welfare for all his devotees and if we observe the deeds and preachings of Adi Shankaracharya, it is he who did this work. In the Linga Puran Purva Bhag Chapter 40 verses 19-25 it has been stated that in Kali-yuga, even Bramhans become critics of the Vedas and other scriptures in order to carry forward their own agenda, in such time, lord Rudra (Nila-Lohita) incarnates as Shankara in order to reestablish the doctrines of the Vedas and other scriptures. The same above Adi Shankaracharya which has been stated in the Purans have been stated in the Shaktisangam Tantra Chinnamasta Khand as well.