Can air conditioning affect transmission?

Defaulting to second gear is "limp mode" and is intended to allow you to drive slowly to the repair shop without needing a tow truck; never at 60 miles per hour. Continued use over 20 - 30 mph is inviting serious engine damage. That is clearly spelled out in the owner's manual. You will need to turn the ignition switch off, then restart the engine to get it out of limp mode so it will start shifting again.

Once the Check Engine light has turned on there will be at least one diagnostic fault code stored in the Engine Computer. Given the circumstances, code P700 can be expected which just means there is a different code in the Transmission Computer. Many auto parts stores will read the codes for you for free but most of them can only access the Engine Computer. You'll most likely have to have a mechanic read the transmission codes.

There are some generalizations you can make by observing when it goes to limp mode. If it does that as soon as you shift to "drive", suspect an input speed sensor or output speed sensor. If it bangs back to second gear during or right after an up-shift, suspect worn clutch plates or internal leakage. The Chrysler DRB3 scanner can display the "clutch volume index", (CVI). That is a set of four numbers showing the volume of fluid it takes to apply each clutch pack. An experienced transmission mechanic can tell how much wear has taken place or how much life is left in each clutch pack by those numbers.

This is only a guess but I would start by looking for loose or corroded ground wires. There are many of them on every vehicle and they share common bolts on the body. A corroded or loose connection will cause current to "back feed" and affect other circuits. In particular, there is a low pressure cutout switch for the air conditioning system that lives near the right hood hinge right by the Transmission Computer. They could share a ground connection. The AC compressor is near the solenoid pack on the transmission and they could share a ground connection.

Another possibility is voltage spikes. Spikes are created by coils of wire when current flow is turned off. That's how ignition coils work and in that case they're desirable. There's a coil of wire in the electrically-operated clutch for the AC compressor and there's a coil of wire in the relay that turns the compressor on and off. Sometimes a voltage spike will cause a computer to lock up or do weird things. You have to turn the ignition switch off and back on to reset them. If this is the problem, you might try unplugging the compressor relay in the fuse box under the hood, then turn the AC system on while driving to see if the transmission keeps on working. The AC won't get cold but that's not important right now. If the transmission still acts up, at least we will know it's not due to voltage spikes from the compressor clutch. Instead of that relay, you can also try unplugging the compressor clutch itself but that's a little harder to get to.

General Motors has had a HUGE problem with voltage spikes from their miserable generators they redesigned starting in the '87 model year. They cause all kinds of problems with computers on those cars and they cause repeat generator failures. The fix to reduce the repeat generator failures is to replace the perfectly good battery if it is more than two years old. As they age, they lose their ability to absorb and dampen those voltage spikes. That's something you might consider too if your battery is getting near its five-year expected life. Chrysler has never had a history of voltage spike / battery / alternator problems but that doesn't mean it can't happen.

I would start by reading the fault codes in the Transmission Computer and resolving those issues first. If an electrical problem is related to a loose or corroded connection, the code(s) should provide a place to start looking. Do not disconnect a battery cable before having the codes read. Doing so will erase any codes, then that valuable information will be lost.