Despite being a tough farmer, mice and rats make me scream like a lil girl

I saw a big gray blur under a hen shed, panicked - peeked another look, and saw it had bunny ears. I instantly was all 'awwwwwwwwww, you're so cute'
In the hen sheds, we use wired bait blocks - kept in specialized plastic cases. Mice & rats can get in and out - but chickens can't reach into them. Usually a rat will eat some bait, get out, and keel over outside the shed, somewhere later. So that sort of setup may work for you - but only if you are 100% happy it is Georgia and Rudy-proof. Have a look at any rural supplies store.
In the egg shed, where we store boxed up eggs, we just use an ultra low frequency pest repeller. It sends out electronic signals which apparently annoys bugs and mice. We were also using a mouse trap with peanut butter - but the lil brats kept eating it and not getting sprung. So we stopped using that