I have spent quite a bit of time on insurance issues and pricing for my rentals and have several observations. The price of insurance is all dependent on the location (mostly by state) of the property. For example, I have properties in TX, FL, AZ and NV. NV is the cheapest, then AZ, then TX and FL is by far the most expensive. It is all based on risk. There are more risks in certain places than others. I pay $500 per year for the same type property that costs me $1,500 per year in FL. And in FL it is very hard to get over 300K in liability limits. There is where you really want an umbrella policy. Most umbrellas require you to have a minimum of 300K primary liablity limits.
You can try to have tenants carry renters insurance too but many will just cancel it and never tell you. It may be in the lease agreement, but good luck. It is like collecting rent from a deadbeat who leaves. Who is the one the ends up being held responsible? You the landlord, always.
In some cases you can also get a commercial policy and that usually comes with higher premiums. The advantage to that is that you can list your LLC as the primary insured. However many personal lines policies for rental properties allow you to add the LLC as an additonal insured. It can get complicated so make sure you use a good agent above and beyond the price. When you need insurance, trust me you need it. I had a tenant sexually assaulted in a rental unit I owned several years ago by someone she had dated. I won't go into all the details but I was held liable and my insurance company had to pay a six figure settlement. This is the society we live in today.....its always someone elses fault. Be smart, insure for as much as you can including an umbrella. If you really think an LLC will protect you from an aggressive lawyer, you have not researched the subject enough.