So, I'm at 53 players under contract.

I made an offer to someone, and also put the order in to cut someone. Will everything happen as expected (player gets cut, then new player may or may not accept contract)?

Or, will it add the new player 1st, cut someone of the sim's choosing, and then also cut the player I ordered to be cut?

Thanks in advance!

Your player will be cut before you sign the other player.

Fwiw .. I always make certain I sign the player first. Then at the next sim make the cut.

That only works if you're going into midweek. If you're going into game week, your team must be down to 53 or the AI will cut someone for you.