Check the dryer timer
The timer sits behind the control knob that you turn to set your dryer's cycle. Its electrical contacts run many of the dryer repair parts, including the heating element and the drive motor. If the electrical contacts don't detach when they should, you may end up with a dryer that just won't stop spinning. Replace the timer if this occurs.
Replace the dryer door switch
Since the dryer doesn't run if the door isn't closed and latched, the dryer door switch is a crucial part. If this part fails, it may inform the dryer that the door is still closed even when you've opened it. If the dryer tub keeps spinning after you've opened the door, look into replacing the dryer door switch.
Check the dryer electronic control board
As your final step of dryer troubleshooting, consider the dryer electronic control board. This complex part is in charge of all your dryer's functions, including heating and spinning. You can spot damage if the part gets burned, but if the fault lies elsewhere in the control board, there may not be any visible signs. Because this is an expensive part, it's best to call in a trained repair technician to replace it.
Sears PartsDirect carries all the parts you need if your RCA dryer keeps spinning when you want it to turn off.