Could Low Oil in my Generator Cause This?

I doubt it would shut off the load, rather than just shut off the engine entirely. An engine running without sufficient oil unloaded is not going to fair much better than one running loaded...probably a little better, but still likely to be damaged before too long if the oil is sufficiently low.

Methinks you may have a problem with your transfer switch or possibly a loose connection between the generator and the transfer switch. If that's the case, it's something you ought to get fixed pretty soon, as these are the sorts of problems that can cause things to get hot enough to smoke or burn.