The reason the A/C doesn't move is because you throttled it. The A/C clutch will kick off to give your engine the needed horsepower to get the vehicle moving.
You will need a scan tool to see what the engine and transmission codes. A cheap scan tool will not work for this. You need one that can connect with the network. I wanted to diagnose to point you in the right direction.
What it sounds like to me is your TTF (transmission temp sensor) is bad and needs to be changed. What happens when it goes bad is you get a check engine light because your transmission module tells the PCM there is something wrong. Your O/D flashing light means there is a fault in your transmission.
What this sensor does is it will tell your transmission when to shift. When your transmission is cold, it will shift early and cause it to slip because the sensor is not actuating the modules in the transmission to work properly. When it warms up, it shifts normal. Sometimes just changing the fluid will solve this, but not always. To change this sensor, you will need to drop the transmission oil pan and remove the filter. It will be the square sensor held in with one bolt on the driver side of the transmission. It is the only one that is accessible without having to take other components off. The rest, you have to take other parts of the transmission off.
If you took it to a shop, the labor guide shows .6 hrs to change it, which is just above 30 minutes. It should be a maximum of 1 hour with clean up, fluid fill, etc. This is for a professional technician to do the work. A novice or DIYer can do it in about a hour and half to two hours, max. That is including jacking it up, etc.
I suggest getting the codes read at you local auto parts store. Some, like Autozone, Advances, Krager, NAPA, and Oreilly's will do this for free. That is to verify you have no engine codes as well.