think, no, make that, “I know”, that you need to have the car’s OBD system checked for stored trouble codes. The inability to drive faster than 5 mph indicates that the car has gone into, “limp home”, mode, which indicates a potentially serious problem. (Serious=very expensive)
My suggestions are as follows:
Check the transmission fluid level, color, and odor.
“Slamming into gears”, is NOT a good sign, but it is possible that this is merely the result of a low level of fluid in the trans.
Be sure to check your Owner’s Manual for the correct procedure for checking the trans fluid, and make sure you determine the correct fluid for this transmission. More than likely, it is one of the iterations of GM’s, “Dexron” fluid, but it is vital to find out for sure before adding any fluid to the trans.
If adding trans fluid doesn’t help, or if the fluid level is normal and the fluid is both red in color and not burnt-smelling, then the problem may not be in the transmission.
In any event, only by having the car’s OBD system checked for stored trouble codes can anyone have an accurate idea of where to begin to look for the source of the problem. If you go to an auto parts store, such as Auto Zone, Advance Auto, O’Reilly, or (possibly) Napa, this will be done, gratis. Then, you can come back to this thread to post the exact codes that were found.