One of my pet peeves also, but there is a simple solution.
Extend your hand with the palm FACE DOWN and grab the bills. Then turn your had over and accept the coins (after you have put the bills in your pocket/wallet).
Stops them in their tracks (almost) every time. I’ve gotten some funny looks as the cashier almost seems to wake up out of a daze they realize they can’t just plop a bunch change on top of the dollar bills in my had.
BTW, I have unsteady hands. Handling both bills and change at the same time is difficult for me (actually, it is a two handed exercise).
Doesn't help much when the ^&%#$&^%# cashier carefully places a stack of coins in the middle of a stack of bills, picks up the stack of bills by the ends, and hands the unstable mess to the customer.
Most such cashiers, FWIW, IMO, seem not to have been born in these United States, nor do they appear as though English were their native language. They also do not appear as though Spanish were their native language.
Actually, I don’t mind the coins placed on top of the bills. It’s easy to slide the coins off into my right hand which goes into my right front pocket, and the bills fold in half and go into the left front pocket.
I use a money clip for bills. The wallet is for credit cards, receipts and other miscellaneous stuff.