Pro/Con? Higher Property Liability vs Commercial Umbrella Policy

[Anna Watkins]

I have several SF rental houses in metro Atlanta/Decatur & middle Tennessee, all with mortgages in my own name, and too many to include in my personal liability umbrella policy.  My insurance agent has suggested that I could increase the liability coverage on each individual house rather than include them all in one commercial umbrella policy -- the total of the premiums would be lower and the deductibles as well.  Seems like a no-brainer, right?

Setting aside the "you should have properties in an LLC" argument (not happening, at least not this year), it seems like higher liability on an individual property would also somewhat insulate the other properties from any single claim -- but am I missing anything that would make me better off with a commercial umbrella instead? Does the umbrella help me more or less if the houses are in different states?

Thanks for your help and experience!