There is no valid reason the clamp should be "slipping off the post" unless there is either something wrong with it, or the battery post is mangled, or both. I think you've already correctly observed the root of the issue. Start there.
Cranking the starter is the sole and single operation with the highest current (amperage) draw out of anything you can do with your car. If either of the battery terminal clamps are making shitty contact, they won't be able to pass that current and you won't be able to run the starter. The other accessories in your car may indeed work just fine if you have a poor battery contact. Your AC blower, lights, radio, etc. draw maybe 8 or 10 amps in total, with everything on full blast. Your starter motor will happily draw hundreds. All of that current needs to pass through the battery terminal clamps first.
Take the clamps off, clean the insides of them with a wire brush (assuming they're the ring shaped type), clean the battery posts of any built up crud as well, reinstall, and ensure that you can firmly screw down both clamps. If the nut on either of them is stripped or they won't tighten down for any reason, either replace the nut or replace the clamps themselves. You can get replacements at any auto parts store and they crimp onto the battery cables.
There's supposed to be a special crimping tool you use for this but no one in human history has ever bought one of those. Rather, we just mash the shit out of the crimp connection with a pair of pliers. It'll be fine. You will have to uncrimp the factory clamps by levering the crimp apart with a big screwdriver or similar. If you're lazy and have enough slack in the cables you can also just cut the old ones off and strip the wire back a little more.
It may also be possible that your battery is on the way out. They don't last forever and they do lose capacity over time. Most auto parts places have a load tester they can hook up to determine if your battery is truly toast, or if it's marginal. That will require you getting the car there first.
Start with the battery terminals.