I have used OM gear for quite some time and have OM2n, 2sp, 4 , 4ti, 20 (G) and 40 models.
As far as I am aware off really is off only with the OM2n (it has the "Off" option on the switch). I have replaced the batteries once in 2 years and about 30 rolls with my OM2 (it lives in my daily pack and gets pretty knocked about), and I did this not because they were flat, but I was going on a trip and thought that I would put new ones in just in case. For reference, the 2sp and 4 models cannot be switched off in any way and I find, using about a roll a week, that I replace batteries about every 6 months. The 4ti is much better on battery consumption and I replace these as a matter of course about each year.
When out of power, the 2sp, 4 and 4ti are able to still fire at 1/60th. This is handy and I was caught once a long way from home without spare batteries and managed to kept on shooting landscapes by applying the sunny f16 rule. There is no way that I know to keep shooting with a 2n without power - with the mirror locked you have to insert new batteries and move the lever to "Check - Reset" before you can resume shooting (unless there is another fix).
Your case does sound odd. The red check light only comes on with my 2n when I move the lever to "check-reset" and goes off again straight away. If your red light is staying on there could be a fault somewhere in the system, but I am not a tech so I can't help there, sorry.
Hope this helps and you have your 2 back on the road sooner rather than later. As an aside, athough the 2n doesn't have the bells and whistles of the other models, I think that it is the smoothest and nicest camera to use out of my OM collection, particularly the buttery film advance - but then each is very pleasurable in its own way. I have just come back from a trip away and I put a few rolls of Ilford Pan F+ though my 2n, with the only problems to exposure, composition, etc caused by me and not a camera malfunction (but life is a learning experience with no set assessment).