I have that exact same cooler. Used it 24/7 on my last recent road trip of 1600 miles and it worked fine. Kept drinks and food at about 40 degrees.
I've had it for about 5 years.
I agree, the adapter gets WAY too hot, and a 10A power supply "should" have run cooler but it probably was only a 5A power supply no matter what it said as someone else pointed out.
Yes, it's a power hog, but the OP has already told us that her budget is a blank so she may not have choices.
I think millions of them were sold and there have only been a few isolated problems, other than their inefficiency.
KrystalsDreamTrip: The AC adapter has to convert AC to DC so a lot of heat is generated in that process. It is safe to run it from a ciggy socket as long as you're not running other things from it. If you're on a tight budget, I would use it until you can do something different. Remember, millions of those were sold and people have used them for a very long time. They are not "ideal", but if your budget cannot afford ideal it may have to do for now.
I still use mine and have no fear of it but I usually use the ciggy adapter. It drops to about 4.5A after it finishes starting up.
I've left it running for days on end that way.
When you can get a small dorm fridge from Walmart that will be better. You might get one of those used pretty cheap in a yard sale or off of offerup or craigslist. They are $80 new at Walmart and work great off your battery and inverter and use less energy than the Igloo cooler.