Cold Transmission Startup Issue

I have a 2002 Buick Century and I have never had any problems with my transmission to date. I recently have encountered an issue when I start my car cold. When I put the car in gear, drive or reverse, you can feel the car try to engage into gear. After I release the brake to start moving the vehicle doesn’t move. In order for the vehicle to move I have to accelerate, but when I do that it jerks violently. It’s almost as if it’s frozen. Granted I do live in Iowa and it’s 5 degrees outside, but I have a hard time imagining that. Once the initial jerk is over, the car runs great. I’ve tried to let the car idle and warm up for about 10 minutes and that doesn’t help at all. I talked to my local parts guy and he thought that my transmission fluid may be getting slushy?? I told him I didn’t know when the last time my car had a transmission fluid/filter change because it was my wife’s car that she inherited from her parents. I talked to a local repair man and he thought that the pump may be getting weak, but I dont know whether to believe him or not. I can’t get the car in to be looked at for 3-4 days so I was hoping for some helpful advice. My father told me to drive the car on the highway for a bit to warm it up and park it in a warm garage and see if that helps. I might try that. Any advice you can give me would be much appreciated.

One more thing to add. I did check my fluids after letting the car idle for about ten minutes and it was ok. I didn’t notice any fluids leaking inside the engine compartment or on the ground. Before I drove the car a longer distance I backed up and went forward a few times to see if I could hear any strange noises and I could not.