Built a 2x2 "ladder", attach it to the joists, and then run 2x2 or 2x4 to span between the 'ladders'.

Otherwise, a picture would help.

Picture 1

I have another picture coming.
I really just need to know if I would be fine spanning the width with 2x4's laying flat, or if I need them standing on edge for support? In order to not have to worry about a retrofit for the bedroom door, I really need them flat, but don't want to risk sagging in the future.

My thought is you should build 3 2x2 ladders. One to the right of the ductwork, one ito the left - between the duct and the pipes, and one left of the I beam. That will give you a support sorta in the middle - then a 2 x 4 laid flat would not sag.

I cut all the 'rungs' on the table saw at once - with the same rip fence setting. Then screwd them together. Just make sure the rungs don't fall on a joist so that it would interfere with attaching it to the joist.

Use a 2x4 16" OC - and attact the drywall.

If the framing is only supporting 1/2" drywall over a 4 ft span, 2x4's turned flat should be more than strong enough.