I carry full coverage on everything. That way if some jackwad runs into my rig my insurance company fixes the car immediately then they get their money back from the fool's insurance company. If you have liability and someone hits you, you have to wait on their insurance company to go through their denial to repair attempts, low buck total offers, and whatnot before getting your ride fixed. It's really not that much more to carry full vs liability especially considering the peace of mind you get.
To add to all that, with the full coverage you also have all the medpay options. I've got enough coverage on my car to cover organ replacements if necessary and in Oklahoma coverage is additive. If you have 100K coverage per car and you have 10 cars, you actually have 1MM coverage for any accident you're in according to my lawyer who has represented us in a couple of accidents.
On storage insurance, done that before too. Can't drive it, but if the house falls on it or if it's stolen you're covered.