Why are 4x4s bad news? If they're crap wood with big knots maybe? You can use a nice piece of 4x4 and let the cross-members in and have the same effect, I think better if done properly, as "sistering" two 2x4s together. Easiest way would be to curf-cut the dado with a skilsaw and knock the waste out with a hammer & chisel. I'd also consider something like countersunk carriage bolts in place of drywall screws for fasteners. Like someone else said screws have very little shear strength.
2x4s or 2x6s alone are likely enough as is a properly built plywood box similar to a kitchen cabinet. The important part is to tie it all together so things remain intact and vertical. Sheathing being the easiest and best way.
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