I don't want to sound contrary, but "normal range" really means very little to me when it comes to thyroid hormone levels, because just being in the "normal range" doesn't mean they're right for you... Some of us need levels higher (or lower) than others.
Many of us (including myself) get rapid heart rate and palps when hypo, so I really can't say a lot without seeing your actual levels, with reference ranges.
If your doctor is not providing you with copies of your labs, you should be aware that if you're in the U.S. s/he is obligated, by law, to provide them upon request.
My TSH, Free T3 and Free T4 were all within the "normal range." Had a cardiologist check me out and stress test, echo and halter came back normal. Now on a two week monitor to rule anything else out. I still get the elevated pulse, so they put me on propanolol, but that made me very depressed. Just was switched to metoprolol which kept me up all night with dreams and palps. I'm so frustrated. Any advice?
What are your actual thyroid hormone levels? They should be testing your for TSH, Free T3 and Free T4. Please post the most current test results, along with reference ranges, so we can get a better idea of what's going on.
It takes 4-6 weeks for a dosage change to take full effect, but you should start seeing some mild changes in the way you feel if your dosage was changed enough.