Electrical problem

Sounds as if low shore power voltage may be an issue. Measure it under load.

Is there a "whole house" inverter on the RV?

Do you also have a surge protector/EMS inline?

Check that voltage under load.

Check your shore power voltage. If it's hot and everyone's running air, that will kill your voltage.

I agree the issue may be on the 120 volt Shore power side of thigns as shore power feeds the conveter. which convets shore power to "Battery" power if the converter is not eating 120vac then it's not working (Technically that is a converter problem but... Well I think you see why it's not) This makes the lights dim and all but A/C fans run slower.

Now the A/C fans and compressor are 120 volt devices but they may rely on 12 volt Relays to activate .

Some differences between a trailer and motor home so I'll do it this way (motor home parts) note a trailer WITH a built in generator qualifies as "motor home" here

One more (optional) part

Suspect list
Shore power
Shore power outelt (park's problem so far)
Shore cord plug
Shore cord outlet (Optional if detachable) and the RV's inlet (LIkewise)
(automatic transfer switch) (Built in generator only)
Main breakers in main breaker panel
Branch breakers
Bus bars

Check to insure breakers are fully seated (they have not tipped foreward and out
And using a flashlight with shore cord UNPLUGED (If possible where you can see it) Check ALL screws and conections

Many RVers I have found seem to have a few screws loose.. One of mine took a #2 square bit and 3 1/2 full turns of the driver before I was happy.

While poking around look for any discolored wires (indicating overheat) sure sign of a bad connection.

I will say again when working in the breaker box UNPLUG and if possible secure the plug where you can see it.

There are also screws that can come lose in the Auto Transfer Switch

Problems I have seen or personally had
Bad park breaker
Bad park outlet
Bad plug connection (mine)
Detachable cord outlet end screws not tight) A brand new 5er and I can tell you he was ..... well you can imagine what he was)
loose screw on inlet (mine again)
loose screws in the breaker box all over the place
Bad connection on an internal plug/socket (mine added you likely do not have an installed hughes Autoformer)

And on the 12 volt side of life.. in addition to dirty battery terminals and assorted other bad connections A BAD WIRE. yup. the wire was bad. (I had and I saw two others)

Not converter as most of your problems are 120 volts AC.
I would be looking at shore power connector or twist lock connector on the rig.
If you have more then one AC unit, try running the other.
You need to verify voltage at pedestal with volt meter. Should be 120/240 for proper 50 amp connection.