I hate to be negative on this one but i think it may be time to retire the old faithfull and get a new printer. You have tried all the standard diagnostics and i think it would indicate to being an issue with the paper sensor that knows how far paper has gone through and thus doesnt know when it is printing or not. Not something that can be replaced to my knowledge.
It would help if we knew what model it was etc. Things to try: 1.) Make sure the batteries are fitting tight between the contacts. 2.) Make sure you are using genuine battery 3.) Ensure the SD card is inserted correctly.
I would think at this stage it would be beyond help, I mean there is a slim chance it would be recoverable but it would be a lot of work and expensive. I would call it quits if it was me. There is a handy guide here if you want to give it a try... Electronics Water Damage