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All automatic transmissions run on fluid pressure
Automatics have Clutches and Bands(brakes) and both are activated and deactivated by fluid pressure, a clutch engages one gear/shaft and the bands hold the other gears from turning, when shifting "gears" clutch is disengaged and another clutch engaged and same for bands(brakes), one is released and another is applied, so only 1 ratio is active in the Planetary gear set at any given time
Ford's new 10-speed automatic has 6 clutch packs!

The pressure comes from the Main Pump, often called Front pump because its at the front of transmission, but its the only pump, lol.

The torque converter is bolted to the engine, and it has two slots or flat spots in its outside casing that mesh with Pump's drive when torque converter slides into transmission
This means pump spins at same RPM as engine, in Park, or any "gear", and that gives transmission fluid pressure when engine is running

You need 120-130psi pressure for Forward gears and 160-175psi pressure for Reverse, higher pressure for Reverse is why many automatic lose Reverse but Forward gears still work, pressure has dropped to low for Reverse
If you increase engine RPMs, so increase Pump RPMs, then pressure will also increase, and gears will often engage when they didn't at idle RPMs

This is the way all automatics worked and still work, not a Ford thing, lol.

The Pressure from the Pump is routed through the Valve body, this is the "brains" of the transmission, in the old days there were ball and spring valves, as pressure increased, Engine and pump RPMs were high in 1st gear, the valve would be pushed open and new path thru Valve body would get the pressure causing shift to 2nd gear, a little more to it than that but thats the basics

In newer transmissions solenoid valves replaced the ball and springs(although some are still used, lol), these solenoids route the pressure thru the Valve Body and can be setup to shift better and smoother than the old ball and spring system.
These are computer controlled solenoids, much better system

BUT...................same problems still happen because automatics still run on pressure, often a valve will stick open or closed, and this effects pressure, or a seal or gasket will start to leak, internally, so pressure is lost.

So your symptom reads like low pressure so slow engagement, if increasing RPMs a bit causes faster engagement then that's the most likely cause

But problem is, where is that pressure being lost at?
Pumps can wear out but really last on the list of problems that occur in Ford transmissions, unless trans was rebuilt a year or two ago, often a rebuilder will either not replace the pump or won't use a Ford Pump, if that happens then trans will only last 1 or 2 years until next rebuild is needed.

Valve body is usually where pressure is lost, and where all the valves and solenoids are, and Valve body can be changed without pulling out the transmission.

I would first see if there are any Transmission Codes, P07XX, not all OBD readers can see these codes
If there are any codes that should point you at possibly a single solenoid or valve issue so you can just replace those

You can go here: https://www.sonnax.com/valve_body_layouts

Then scroll down to FORD 4R/5R transmission and select that, this shows you the Valve Body layout and the valves and seals involved, any of which could be where pressure is being lost or mis-routed
Valve body even "looks like" a cross section of a "brain", lol
Engineers have done all the hard work in that lay out, all we need to do is keep it working

 

 

Maybe EPC solenoid, it controls line pressure in trans

Sprang clutch, AKA one-way clutch or forward clutch, but this requires full disassembly to change