When replacing battery on 93 buick, starter engages

Cerainly - only way to know for sure is to to do a battery load test. However, replacing the starter without the correct shims can also cause this problem.

-the wires are not correctly re-installed for the starter.
- u may need wiring diagram
Did you take pictures before you took it apart, or did you tag wires before unconnecting them.

if your battery is low on charge or you have to jump it off it may already be engaged and stuck in flywheel try turning motor by hand back and forth listen if you hear it pop out then try to start with good battery if doesnt work replace starter it could be the selinoid switch but you have to pull starter anyway so just replace all of it

could be a problem starter relay or starter solenoid
if the battery lead goes straight to the starter , pull the starter and have it bench tested

Without hearing the sound it makes - it's kind of hard to diagnose.

The engine starting process consists of turning the key to start, the starter motor's rotor moving forward to engage the engine's flywheel, and then the starter motor 's "rotor" spinning until you move the key from "start" to "run". You can hear the starter motor spinning - but one usually can't hear the rotor moving forward to engage the flywheel. As a result, you hear the odd spinning sound - which is the starter rotor spinning. The engine never actually turns over because the rotor never engages the fly wheel. If this is the case, then it is very likely that your starter needs to be checked and / or replaced.

Good luck!

Test the battery, if it is ok replace the starter solenoid, they click when the contacts get burnt in them and won't connect to engage the starter.

no you have a bad connection somewhere causing the problem. clean and tighten the battery terminals and make sure they are tight as they would cause what is going on with your vehicle

You still have a poor connection between the battery and the starter. Or a defective battery. (the headlights, if turned on will nearly go out when the key is turned to start)

REMOVE BATTERY TERMINALS AND CLEAN REINSTALL AND TRY TO BOOST AGAIN BEFORE YOU SPEND OUT ON STARTER OR MORE EXPENSIVE ITEMS

Recheck the battery, the starter doesn't care if your battery gives 12 volts, it needs CCA (Cold Cranking Amps),  Autozone will test this for free where I live, most likely where you live too.