I keep getting paper jam on my hp office jet 5610

Hello there
You may not see a paper jam.

This may work for you, it has worked for many here I given this advice so gave it a try it could work for you. Generally when this happens, something has fallen into the printer, or a piece of paper has been jammed in there.

First step, remove the ink cartridge, and turn the printer upside down and gently shake it, see if anything comes out.

If nothing goes, or you can't hear anything rattling around, get a piece of cardstock paper (60lb or heavier) and gently push this through the printer, to see if it will help push out what might be stuck in there.
And gave the rode that the ink slides on a good clean and maybe a bit oil on cloth and wipe the rod with it.

Please come back and just gave a vote, it will gave me some satisfaction to know am helping some one so I will keep on helping.
All The Best: Pineworks