I use the Hydrofarm Jump Start Thermometer. It's around $30 and you can get it off Amazon with Prime shipping. https://www.amazon.com/dp/B000NZZG3S..._mCPNwbCQ4XBRT (If you do not have Prime membership, sign up for the 30-day free trial and then cancel it when you're all done.) I've been using it for a year and while it's not perfect, it works well for what you pay for.
Contrary to what someone else said, you do need around a 90-93 degree hot spot in the cage. 80-85 ambient temperature is good and he'll be fine with that until you get the thermostat.
The loose skin and the cloudy eye caps were just stuck shed. The humidity in his cage before you got him was too low. If it happens again or there is still some shed left, soak him in slightly warm water and boost the humidity in his tub until it's gone.
0.1 Super pastel