Does it work in simulation?
Are all 4th axis movements causing the y axis to move or just the engraving tool path? That could help narrow it down to a config issue or a tool path setup.
I would double check the tool path, especially the section where you set up the multi axis. When I first started this area vexed me while I sorted it out.
In the configuration of your nc double check how you setup the 4th axis.
It could help if you post some screen shots.
This project doesn’t have any engraving in it but it’s the only 4th project I have available ATM. It might help to plunk through there to see how I setup the job.