Sounds exactly like a torn separator gasket, which is addressed by Ford's TSB for your transmission and a few odd parts - Prolly $150 of parts and 3-4 hours under-car - no need for the tranny to come out. -- the Trans-go shift kit should also address this issue (new gasket), though I'd recommend sticking in a new EPC Solenoid while you're on the bench.
I'd suggest pulling the pan and looking for busted band parts or massive amounts of friction material in the pan -- If you don't have broken parts and can assume that the bands are OK, then (since gears are engaging for the most part) look at the sep gasket and EPC solenoid performance for the issues you describe ...
Here's a photo of fairly typical sep gasket failure for the 5r55e (photo source
) You can see the blown-out area at the Reverse servo and also near one of the shift solenoids... Pressure lost to these leaks will keep the solenoids from engaging exactly as you describe.
Only a few bolts to get you this far into it -- and give you a good understanding of what you've got.
Even if all you do is pull the pan and look -- you'll have drained out a fair amount of ATF (~5qts) so if you end-up pulling the tranny, it'll be a bit less messy.