Thanks for the service information. My watch guy says he has experience with Accutrons. I will check on a price but I doubt I will spend for a service. I wear my vintage watches maybe once or twice a month for fun. My Deep Sea is running unpredictably on the 344. Will run perfect time for half a day and then 1-2 minutes fast the rest of the day. I will try the Accucell on this one and see what happens. My other dress Accutron runs pretty well on the 344. On or off the wrist it runs 20 sec per day SLOW. I would think Accutrons were designed to do better but this is at least usable and acceptable to me. I wouldn't think the Accucel would help a slow running watch? This one might be better suited to a service? My Accucells haven't arrived yet. I will update what happens with the new battery but I certainly understand and agree that lack of service will muddle the results.
If the index mechanism was set for the Accucell (or original battery) and the 344 is put in it, it'll run erratically fast depending on position. Sounds like it could be the case with yours.
I've read that unlike a mechanical movement, that will refuse to run when the oil is completely shot, the Accutron movement will soldier on wrecking itself. If it starts hanging up with the Accucell and doesn't keep time consistently, a service would help it greatly. I wore mine for a few months when I bought it until it stopped running, and in that time I realized it was a keeper and saved up to get it serviced. Hopefully this one endears itself to you.
As for accuracy, my 1976 Accutron with the 218 movement keeps within -1 sec/day on the wrist or in the box. It took me a fair bit of trial and error with the index mechanism to get it to run on the modern battery, but they can do it. It's in the dial up, 12 o'clock angled down position right now to gain back a few seconds after holding steady at -5 sec total for the past week.
I can't remember which one of these recent Accutron threads I mentioned it on, but you can use the Accucell and the 344 battery to phase the index mechanism for the 344 if you really want to.
Good luck with it. Really is a beautiful watch.