If you choose to reclass into a different MOS, are you still able to pick your next duty station when signing your reenlistment contract?

You can do whatever is stated in your contract. There is no "blanket" statement that answers your question.

No, you will enter AIT as a MOS-T and be given your follow on assignment close to the end of your training. You are needs of the Army

No, you reenlist for training not a duty assignment. You don't even reenlist for a training date, just the training.
You will attend training in a TDY enroute status and you can pick up to four preferences for your follow on assignment. Needs of the Army trumps all and if you are EFMP 90% chance your follow on assignment will be Korea

This is a conversation to have with the individual who monitors manpower for your new MOS. It is most likely the same person who processed the lateral move to begin with. Ultimately you are nick the new guy in the MOS and have little pull but you can try to communicate with the monitor and see what is available. I lateral moved coming off of Embassy duty and was able to communicate my preference via my 1st Sgt and it worked out but it is not a given or guarantee. At the end of the day you go where you are sent/needed

Nope. You are needs of the Army. There is nothing saying that the duty station you select will need your MOS at the time you get ready to PCS.

So I reclassed to 92y and I’m leaving for ait In july. Retention said my orders said fort drum after Ait. Is it possible that it will change after Ait ?

Your can't choose your duty station when you sign your contract for reclass, but once your paperwork is received, you can reach out to the branch manager of your gaining MOS. They might be able to accommodate a request for a highly qualified soldier, especially if you've already found vacancies via ACTnow.