First, if you haven't -- change your transmission fluid. Get it flushed and refill the whole thing with a good quality fluid, preferably sythetic.
If that doesn't help, then you probably need some parts rebuilt/replaced. That would primarily be (if you're lucky) the servo's themselves, or a few hard parts like solenoids. All of these can be done with the tranny in the vehicle.
If you're very unlucky, then your forward clutch piston is sticking or the pressure to it is low and you may require a more extensive rebuild of the transmission to replace either the forward clutch piston seals, the clutch plates, or both.
Your symptoms definitely sound like your forward clutch won't engage until it's hit with a certain increase in pressure -- like it's sticking. From what I know of the transmissions operation, this can come from pressure problems to the clutch or the clutch itself. Hence the thought that it's the forward clutch, the solenoid or valve which sends pressure to it, or the EPC solenoid that modulates shift pressure.
It doesn't sound like the differential to me.
I have a thread on rebuilding the 5R44E if you look it up you can see what some of these parts look like that I took when I rebuilt mine.
If your fluid is dead, sometimes just new fluid can do wonders and it's probably the cheapest thing you can do to try to fix it. But getting a professional diagnoses rather than one by "remote control" on the forum is my recommendation.