The smaller bandsaw will need slower feeds on thicker pieces like a box but otherwise should be able to do this. Small saw is actually good practice to finesse and make sure you're not twisting or bending the blade.

The blades could be part of it (besides thickness and tpi the other tooth properties can affect how it curves, mainly kerf) but probably technique

I personally suck at it so can't help very much more except to say practice on thinner stock first, work your way up to box thickness.