i have one of the same transmitter, if you dont connect a propper antenna, the out put transistor gets very hot, even chances to get damaged, but the point you mentioned, it only gets very hot when operated at 1W is not clear.
you better conform you use a propper antenna or a dummy load, then see the range with both 1W and 10W out puts.
about power supply getting hot... are you sure it only happens when operated at 1W? may be some thing wrong on the display control in the front.
also keep in your mind operating such transmitters are illegal in most of the countries..