>>Just write them a check for the difference<<
Send an amended return along with the check else they won’t properly credit your account.
Even if the first one isn’t processed, amending is the safest as you will be able to prove you filed correctly when you knew something changed.
Yes, with an amended form 1040, with the check and a letter stating what it’s for attached. The point is to get them the money as fast as possible to head off any fines or interest.