We've, actually, had numerous members, both male and female, who were born without a thyroid gland. Thyroid hormones are necessary for growth and development, both, as a fetus and as children. It's good that your mother realized, early on, that you had an issue.
It's quite normal for those of us with thyroid conditions to be intolerant to extreme temperatures, particularly, intolerant to cold, when we have hypothyroidism. Typically, if medications are adjusted properly, the cold intolerance is not so bad.
I'd have to ask what your actual thyroid hormone levels are. If you could post your most recent lab test results, we could better assess your situation. You should be getting tested, regularly, for Free T4, Free T3 and TSH. 300 mcg of Levothyroxine is a pretty hefty dose that we don't see very often, even in one with no thyroid...