I’m probably not the best person to answer this, but I’m going to try anyway. For the longest time, I had trouble with my left hand cramping up on me when I played mando, and I think it was caused by exactly what your describing - trying to support the weight of the instrument with my left hand while I fretted. Doing this put my hand in an awkward position that became painful after only a short while. From your description it sounds like your E-string problem may have the same underlying cause.
I minimized my problem by getting a strap adjusted to the right length for me, and always using it even when sitting down, but even now I have to be aware of slipping back into the bad habit of supporting the mando with my left hand.
With your round backed mandolin, though, I’m not sure of the best way to hold it. Seems like it would be even more unstable than my f-style. Maybe a more accomplished player will come by and answer that for us.