car coolers for travel plug in?

In the summer months, I keep a good quality cooler in the back of the car. Most of my shopping is between 45 minutes and an hour away. (I live in a very remote area.)
sometimes I throw in a blue ice block. That has worked well for me, but summer highs here are usually in the upper 80s at worst.

do you leave it plugged in when not in use, to me that is what makes the convenience inconvenient (and if I forget to unplug as the manual says to do what happens, dead battery?)
80's is plenty hot enough to ruin a grocery trip, the Blue Bloks sounds good

Sorry, I should have more clearly and fully described the kind of "cooler" I have. I was speaking of an Igloo brand cooler that I have had for about 25 years. Maybe 45 or 50 qt capacity. No wheels or other frills. It has thick walls filled with insulation, not simple voids. When new, the sticker on it claimed it would keep food cold 5 days in 90* temperatures. I bought 2 of them for camping back in the days when I still did tents. No plug-in of any kind. Being well-insulated and with the help of the blue ice block, it has been adequate for getting frozen food home. The blue ice may actually be overkill. I figure it is extra insurance to keep the inside of the empty cooler cool in the hot car sitting in summer sun while I am in the grocery store.
My inclination is to keep things simple where possible. I have no experience with portable plug-in coolers.