Only thing I would add is to replace the coolant lost with water initially - this saves cost of more coolant - UNTIL YOU ARE ABSOLUTELY SURE THIS SOLVED THE PROBLEM.
If the new tstat did fix the problem, then drain a bit (your choice of methods) and add appropriate amount of coolant. I have heard a lot of variance in temperatures shown on the gauge that I think are related to the mix of coolants.
1991 350 SDL (200K and she ain't bent, yet)
former 2002 E320 4Matic Wagon - good car
former 1985 300 CD - great car
former 1981 300 TD - good car
former 1972 280 SEL - not so good car
a couple of those diesel Rabbits ...40-45 mpg