If I reclassify to 88N, does my new contract start once I complete AIT and clear my old unit?

Reclassing does not necessarily come with a new contract. Your ETS should be the same unless you're reupping or extending.

Your new contract starts the day you sign it. When you reenlist, you are technically discharged your new commitment begins from that day.

I’m not a reenlistment NCO so I’m guessing here, thinking your new contract starts the minute your raise your hand to re-up and has nothing to do with the school itself. Like I said I’m taking my best guess on this one.