If you can get up $10K more or so, then Alabama could work. It’s a well known name.
South Dakota State has MechE. It shows $25.3K COA . You’d get $4K merit so low 20s. I know, middle of nowhere but it’s ABET accredited in MechE. Slippery Rock in PA is also low cost for MechE.
These are the kind of schools to look at for affordability.
But don’t forget, there is travel expense too etc.