Welcome to the Nation!
Yes, break barrels can move (if they don't have engineering to deter that) and they especially develop "droop." The latter is addressed by buying a shimmed scope mount to compensate. The former requires that you check the mounting screws regularly and keep them tight so the barrel will be less likely to move.
I got a Crosman Nitro Venom Dusk (.177) that is very accurate (killed squirrels out to 25 yards with it). It's a solid performer with a sucky trigger (but it works)--I've been too lazy (and cheap) to do a trigger tune so far. Each entry level gun is a dice roll; but some are really good, right out of the box.
BTW, you need to shoot a tin or two (1,000 pellets) to really break it in before you get too concerned about performance. Then, there's cleaning the barrel and regular maintenance, too.
Best of luck,
Arch_E