Since the warning light is present, the first thing I would do is have the computer scanned for diagnostic trouble codes. It could be on for a few different reasons and there may be codes that are relevant to the transmission issue.
Now, you mentioned the transmission fluid is full. Is it clean? Does it have a burnt odor to it? Are you checking it at operating temp with the engine running?
Next, if I understand correctly, the transmission seems to go into neutral, the engine revs and then it goes into gear. That could be related to transmission line pressure. If it is low, it can cause this. There were some issues where the transmission cooler required back flushing due to partial blockages. Also, there is a line pressure solenoid that can cause this.
What I need you to do first is scan the computer for codes. Here is a quick video showing how it is done:
I realize most people don't own a scanner, but often times, a parts store will do it free of charge. This needs done. Otherwise, it becomes a guessing game. I don't want you to spend time checking things which are not related. I hope you understand.
Let me know what you find.
Thursday, April 9th, 2020 AT 7:41 AM