Yes, Karna was a student of Drona. But he also learned from Krpa and Rāma Jāmadagnya (Parashurama). This is confirmed in the Vana parva of Mahabharata.
And seeing that in process of time his son had grown up, Adhiratha sent him to the city named after the elephant. And there Karna put up with Drona, for the purpose of learning arms. And that powerful youth contracted a friendship with Duryodhana. And having acquired all the four kinds of weapons from Drona, Kripa, and Rama, he became famous in the world as a mighty bowman.
Your second question -
But all normal stories say that Drona declined to teach Karna, and Rama was his teacher.
Drona declines to teach Brahmastra to Karna. Hence Karna approaches Rama.
Source from Mahabharata
Beholding that Dhananjaya was superior to every one in the science of weapons, Karna. one day approached Drona in private and said these words unto him, 'I desire to be acquainted with the Brahma weapon, with all its mantras and the power of withdrawing it, for I desire to fight Arjuna. Without doubt, the affection thou bearest to every one of thy pupils is equal to what thou bearest to thy own son. I pray that all the masters of the science of weapons may, through thy grace, regard me as one accomplished in weapons!' Thus addressed by him, Drona, from partiality for Phalguna, as also from his knowledge of the wickedness of Karna, said, 'None but a Brahmana, who has duly observed all vows, should be acquainted with the Brahma weapon, or a Kshatriya that has practised austere penances, and no other.' When Drona had answered thus, Karna, having worshipped him, obtained his leave, and proceeded without delay to Rama then residing on the Mahendra mountains.
Rama gives instruction about Brahmastra to Karna.
Narada said, 'That tiger of Bhrigu's race (viz., Rama), was well-pleased with the might of Karna's arms, his affection (for him), his self-restraint, and the services he did unto his preceptor. Observant of ascetic penances, Rama cheerfully communicated, with due forms, unto his penance-observing disciple, everything about the Brahma weapon with the mantras for withdrawing it.