Buying Liability insurance along with property insurance.. Question

[Anna Sam]

I am currently buying Liability insurance along with property insurance when I buy a new property. I also have a general liability policy for my business too. I own a few properties within the LLC.

If I skip the Liability insurance while buying property insurance I dont really save much so I add that too usually.

My question is ... If I have multiple policies for liability insurance (genl liability for my LLC and general liability + property insurance for each property) would there be an issue with 2 insurance companies fighting to not pay when I need it.? i heard something like that and wanted to check.