A/C stopped; added some 134a, now clutch cycling

Hello, the clutch should be cycling about every 8 seconds. That is a normal operation. I'm assuming you're monitoring the pressure with the little gauge on the refill can. That only measures the low side pressure. The fact that the system was low enough on Freon that it stopped working means there is a leak in the system. While adding refrigerant to the system might get the a/c to work temporarily, there is air in the system now. as well as moisture. The correct way to restore the system is to find the leak, replace the Receiver Dryer (the filter for the system), then vacuum the system for 30min until all the air and moisture is out of the system, then refill it with the correct amount of R134A. An average of 40PSI on the Low-pressure side is normal but the air that is in the system now is throwing off the reading.

https://www.2carpros.com/articles/re-charge-an-air-conditioner-system

https://www.2carpros.com/articles/air-conditioner-leak-detection

If there is still a little pressure in the system, I would just use a spray bottle and a little soapy water while it's running. Spray the connections, the condenser on the front near the radiator, and the fill valves (they leak a lot more than people think). The leak will bubble up quick, you are going to have to have the system vacuum down anyway after the repair and refill. It's quicker than putting in dye and wasting money on another can of Freon.

Looks like there might be too much air in the system, air is much more compressible than Freon is. If it's cycling that fast it's not building enough pressure to do anything really, did you try spraying the whole front condenser?

You can try putting some more Freon in, but you have a big leak if it's empty overnight so I wouldn't waste the money on Freon that's just going to leak right out. It will need to go to a shop for the system to be vacuumed down and recharged anyway. What you can do is have them call you before they do any repairs. If they call after they find the leak, then you will at least know how much the repair will be. If it's a faulty compressor, it can be quite expensive. With a failed compressor the whole system needs to be cleaned out, the receiver dryer needs to be replaced, the lines need to be cleaned just in case there's particles from the failed compressor, etc.
Thats what I would do so you don't go to pick it up and get a huge bill you're not prepared for. Especially with a big leak, it might be the condenser on the front of the vehicle, and those can be costly as well. So have them call you after they find a leak and give you an estimate for the word. Just to cover yourself.
Also take it to a shop you trust. I hear too many stories of people being taken advantage of and it's just wrong. Hope it all goes well for you. And let us know what you find.

Me too, that's the worst place. Thats why i'd have them call you first, the bill for that might be too much. I'll take a look and see what needs to come out if the evaporator is leaking,

What size engine is in this, and is this a pickup or van? Dodge has listed a Ram Van and Ram Wagon as well.

Anyway, it looks like the whole dash has to come out to get the HVAC box/housing out. It's a big job.

It's no bother at all. If you shake the bottle of soapy water up and have enough dish soap in there, you'll find a leak anywhere because the dish soap sticks to components. I've found a lot of leaks at the fill ports before. It's actually the mystery spot technicians miss all the time because they have both gauges set on the ports while trying to find a leak. Then they refill the system, put the caps on, and ship the vehicle, but it still comes back with a leak due to those ports. I have not changed out one of the valves, usually just that section of line will be replaced, I don't think they are regular Schrader valves like in a tire. Because of the temperatures and pressures, they have to be able to take. But use really soapy water and shake it. If there's a leak it will bubble out. I learned about these valves by chance one day, I happened to be working in a quiet place and I took the cap off and heard a hiss. I sprayed the port and sure enough it was leaking like crazy. But you will hear a little hiss when taking those refill cans off because there is still a little Freon on the end of the line. I'm not sure if you can buy just the valve part, I've never seen them for sale, did you try spraying the pressure switches too? They will sometimes leak right through the sensor. But definitely do some more investigation of the valve ports. They are highly overlooked. Just remember the trick is extremely soapy water and use dish soap. You can also use it for tire leaks. Works great.