It’s not AA, but I think you’d enjoy this:
Take a Convoy S5 (the smallest of the S series) for $13.80, replace the reflector with a 60° TIR optic (I realize that’s a 5-pack, but it’s hard to find these in singles) for $1.61. The optic perfects the Cree beam & this one makes a nice flood for any use within loud-talking distance.
You’ll need to inform yourself as to the proper care and handling of LiIons, but “with great power comes great responsibility.” I take all mine from “dead” laptop packs, but that takes a higher level of commitment than some are willing to invest. We can’t escape the hysterical hype even here, but there is some excellent information to be had (link is to one good example, keep searching for more), if you keep a level head. My cost, $0.00, you choose. A reasonably-well-reviewed “boughten” version is here, but I can’t recommend what I’ve never seen. $4.61 each.
A decent USB charger for 18650s can be had for $4.87, and it both charges via USB and provides USB power to your (e.g.) cellphone, etc.
So how does this fit into your $20 budget?
Convoy S5: $13.80
Beam-shaping Optic: $1.61 (technically $0.32, but I can’t find them individually)
Charger: $4.87
Batteries if purchased outright: $4.61
Total: $23.60 each… 18% over budget isn’t good engineering, but it works for me. (Full disclosure, I built mine from parts, so it cost a lot less, but this is what I ended up with) It also makes a wonderful fill light when you need it.