Just do a ACH balance transfer, move your bills, then close the account. No need to be driving a paper check around.

If you'll think billing problems will occur, break the transfer into two chunks and keep both accounts open for two months or so until satisfied.

For large amounts of money, I think the best way is to open your new account, wire transfer (not ACH or check) the money to your new account, confirm that the funds are received, then close the old account. This will likely cost ~$25 depending on your old bank but it avoids the hold periods you'll otherwise get with a check (even a cashier's check) or ACH transfer, as well as issues with moving physical cash.

You don't need to worry about AML/KYC laws, those are the bank's problem. In fact, using smaller transfers to avoid hitting the $10k threshold is illegal.

The only significance of $10,000 is that it’s the amount that triggers reporting. Such transactions aren’t illegal or even suspicious. They’ll write you a check for the full amount and the other bank will accept it. There might be a hold on the funds for several days until the check clears. As another commenter mentions, you can open the new account first and do an electronic transfer which will make the funds available sooner.

depending on your original account, if it has a minimum balance to avoid fees you might want to leave the minimum balance in there and transfer the rest first. Then close the account officially and get a check for the remainder. the reason is that if you transfer it all and it goes under a minimum deposit, you could get a fee and then you are overdrawn, generating more fees and more overdraw. Generally just taking all money out of an account doesn't automatically close it.