Hesitation/Stutter

If I’m driving my 2000 Dodge Durango uphill at speeds beyond 50 mph, and applying the gas gradually, the engine will drastically stutter or hesitate. In addition, if I “punch it” like I’m trying to pass another car, the engine revs extremely high - what we often called redlining and the stutters actually shake the vehicle. My hometown mechanic said my transmission was going bad, so I took it to a highly reputable transmission shop. They had it overnight for driving and testing and said there was nothing wrong with the transmission. Other speculations include notions of a vacuum leak and/or a fuel injection problem. Any ideas are appreciated.

Is the ‘Check Engine’ light on? Does it do on when the stuttering happens?

Get the Dodge scanned for codes first. AutoZone will do this for you free.
If there are no codes present I might be leaning towards either a fuel flow problem (clogged filter or failing pump), vacuum leak, or an air leak in the intake tract.

A vacuum gauge can be used to check for an intake leak at idle.
If the fuel filter is removed you might allow it to dry for an hour followed by blowing through it. One should be able to blow freely through it. This should not be done while the filter is wet as it skews the results.
An air leak in the intake tract will affect the MAF sensor. A problem here is usually minor in nature.

None of that may be the problem but it’s what I would consider at this point.

Okay, now that you mention it…there’s additional (possibly unrelated) issues here. Occasionally and randomly, as the Durango is being driven, the open door tone goes off several times, the speedometer and RPM needles fly back and forth, all the warning lights come on. Also, the radio has quit working. The only way to get the radio to work is to disconnect the battery cable for a minute and reconnect. Then, the radio works for an uncertain amount of time.