New Yorker transmission trouble

Hello

I am a senior citizen. I need to have my transmission looked at, but
don't want to get ripped off by a transmission shop either doing
damage by 'testing' the car or by partially dismantling the tranny.
Then they tell you it is broken and needs a new transmission, and they
regret that they can't put it back the way it was when you drove in.

It's an older Canadian car, a 1993 Chrysler New Yorker Fifth Avenue.
It has a 3.8 Lit 6-cyl, automatic with a/c (that doesn't work
anymore).

When it is cold first thing in the morning, I start the engine, put it
in reverse then wait a half minute or so before it starts to back up.
If I rev the engine, nothing happens. This is starting to get worse,
but it could be just because winter is starting to happen. I live
just north of Toronto, Ontario, Canada.

Lately, it is starting to shift erratically. Sometimes it seems it is
in a lower gear, then suddenly shifts to highway cruising gear.

I know nothing about transmissions. I have had the car about a year
now, and keep checking the fluid, but it is always ok, and doesn't
smell or look abnormal.

So...to keep from being ripped off...can the transmission be checked
without any problems, and left the way it was, by a reputable shop?
What could the possible problem be? Could it be just an adjustment
necessary?

Any help would be MUCH appreciated.

Thank you, and Merry Christmas!