I think what you are doing is fine, though I am concerned the temperature of 90 °F may be too much. You could reduce the wattage of the bulb and use a heating pad at the same time if needed. I wouldn't let it drop below 70 during the day. Mid to upper 60's is fine at night.

White's are found over most of eastern Australia and most of the island of New Guinea. There is also introduced population is Florida and New Zealand. They are found in many different ecosystems, so they can deal very well with different conditions, such as temperature and humidity. So, try not to worry to much.