Gallium is fine. The thermometers will be calibrated for this metal. Gallium is actually a pretty interesting metal. It has such a low melting point that it will melt in your hand.
Falling asleep while taking the temperature will not hurt anything. Glass thermometers should be held under the arm for about 10 minutes anyway to get a more accurate reading.
Your first reading is a little low. But a series is really needed to get a better idea of thyroid function since the temperature can fluctuate.