I use cauls and pipe clamps, so at worst one side will be flat. The other side may have a seam with 1/64” difference in height. Just enough to be felt by the hand.
I wipe up any squeeze out with a wet cloth. I use a scraper with a carbide edge for any touch up.
This is exactly the way I do clean up. You will hear people say never clean up wet glue glue using a wet rag b/c the residue will get in the grain and interfere with finishing. I can say this is not true if it’s done properly. They say you wait till it’s rubbery or dry. Who sets a timer, and letting it dry is asking for a tear out disaster. That said, with panels, if you’re thicknessing after glue up it’s not an issue. In fact you don’t even need to clean anything at all.
If I could say something about panels, panels can be stressful especially for a beginner. One of the biggest mistakes is milling boards to size prior to glue up. That said, if you are forced to buy S4S lumber you have no choice. That forces them to get a perfect joint, so they have to resort to alignment aids such as dowels, biscuits or heaven forbid, they buy a Domino.
I keep panels thick and mill to final thickness after glue up. If the board are straight I will even glue up rough lumber. A router flattening jig is fine, often I can flatten one side with hand planes. I have a 20” planer, anything wider than that I use a drum sander with 50 or even 36 grit sandpaper.
You are cleaning up squeeze out exactly how I do it, except I scrape it off first, then clean up with a wet rag (do not use paper towels).
I’ve never worried about what little dried glue may remain or missed in either a planer or drum sander.