Hypothyroidism can cause all of those things to happen, in spite of the "not very high" TSH... Your doctor is horribly wrong in saying that your TSH would have to be "way out of range" to be symptomatic... Many people are symptomatic with TSH "in range". The reason for this is because TSH is a pituitary hormone, not a thyroid hormone and it does not correlate with symptoms at all.
Free T4 and Free T3 are the actual thyroid hormones, with Free T3 being the one that correlates with symptoms and one can even be symptomatic if FT3 and FT4 are "in range", if they are too low in the ranges.
When I was really hypo, my heart rate was dropping into the low 30's when I slept... I also had some rapid heart rates and palps, along the way, as well.
When you get your test results, please post them, with the reference ranges and we'll be able to better advise you.