It is always a good idea to service the trans. There isn't anything you could do inside the pan that would cause an issue unless you installed the sump filter seal incorrectly. Don't worry about that.
What you are feeling in the trans when the truck is cold happens to slot of the RFE's and even the RH/RE transmissions.
What is likely happening is the fluid in the torque converter is draining while the engine is off and the truck is sitting. When you start the engine and put the shifter in drive the pump starts to fill the converter with fluid. When you press on the accelerator the converter is still being filled with fluid and it feels like the trans is slipping. There is a check valve in the trans cooler return hose that is supposed to prevent torque converter fluid drainback. If this has been happening for quite some time I suspect the drainback valve has failed and is allowing the fluid in the converter to drain out while sitting.
When you service the trans, pay close attention to how the sump filter seal is positioned into the case. You can pry the old one out with a flat blade screwdriver. To install the new seal, use a ⅝" deep socket with a 6" extension and use your hammer to drive the new seal into the case. It doesn't take much force from the hammer to install the seal. Just a couple of nice, light taps and it will go in. You need to make sure the seal is flush against the case and that it is not loose.
For the spin-on cooler return filter, it's just like the oil filter for the engine. Use a large pair of channel-lock pliers to break it loose or a small oil filter wrench. When you install the new cooler return filter make sure you don't cross the threads because they are plastic and it can happen if your not paying attention. It's actually very simple.
Also, don't use a gasket for the trans pan. It is designed to be installed with ATF/RTV. The stuff from your local Mopar dealer works the best. ATF/RTV is designed to seal even if it comes in contact with trans fluid.
If you have My questions or need any help just let me know.