Hesitates a few minutes before shifting into reverse and then drive. I have serviced this trans according to service manual. Car has 173K miles. Now it is getting worse, RPM at 3k at 60mph. As it warms up it clears back to normal around 2k at 60.

Usually bad "solenoid pack" inside automatic trans. Although, you usually also get a slight hesitation followed by a "clunk" when switching from forward to reverse. Also possibly computer programming error. If it has "adaptive" shifting (i.e. "computer learns your driving pattern and shifts accordingly"), you can have Ford change the programming to a standard shift pattern. (It's a wonderful difference in transmission performance).

Service the transmission for the shifting issue. However going from auto to manual is an electrical switch. Car wont shift till car is at high enough rpm level.

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did you forget the 50k mile service? full flush and filter.???
ooops......... zero service equals LOW LIFE> EVERY TIME./
check fuses?
check levels per operator guide, (hint dont guess)
fails hot and cold engine?
fails all gears, 1,2,L, Drive? and Reverse?

It has a worn or broken rear engine mount most of the torque is absorbed in the front mounts and hanger mounts except when it is shifted into reverse.

Could be a long list of things.
Its too late for a filter change. Doing that now would probably make matters worse.
The trans uses electric solenoids and a pump to apply pressure to the clutches. So either could be faulty given the age and mileage.
I'm not sure if the shifting problem is connected to the shift indicator lights problem.
You may need to take it to a trans shop to find out for sure. And they will not want to touch it with that many miles on it.
A regular mechanic might be able to do a pressure test on it to at least find out if it is the pump or not.

Taker it in for a tranny service.

that delayed reaction may be the transmission slipping. with those many miles it could be the transmission going bad

How many miles? Have you maintained, serviced trans in past or just waited for this day to wonder what happened?

i would start with an oil change for the transmission this should help, the oil should be changed about every 30,000 miles, if that doesnt help try some trans x, it helps automatic transmissions shift better

is the trand fluid full? has the trans fluid and filter been changed every 60,000 miles? if not service the trans and retest, many times a fluid and filter change cures this problem, if it doesn't u will need to have the transmission serviced completely