In the beginning of Hashi some swing between hypo and hyper, eventually going into a declining hypo state - which is what Hashimoto is (hypO).
Peolpe can get heart palps and high BP from Hypo also - their heart is working to hard to keep up. This is what I get when meds are to low. When hypo I really cant exercise safely - heart races and body hurts for days after. Up my thyroid med and its a lot better. Bottom of the T4 range is the same as hypo for me - some doctors refuse to beleive this.
I have hashimoto's, but swung into a hyperthyroid state followed by hypo. I now am off medication. I started exercising right before I went hyper. I'm not sure it triggered it or it is coincidence. Now though, like you, I can barely do anything without my heart racing. I can do minor activity (such as running up the steps.. sad but true). I guess at the beginning of hashi's it is common to fluctuate back and forth between hyper and hypo until the thyroid dies enough to become completely hypo. I had a horrible year of it doing that, and now it has settled down to where I don't have the extreme panic and anxiety. I also have less palps, and very few episodes of heart racing. I don't think I could run a race yet, but maybe I'll get there. I feel your pain though =(. I always thought maybe the exercise made me healthier and triggered my thyroid to work more, which made me hyper. I don't know though. I know once you get medication optimal and your thyroid steadies itself, you will feel much better. Good LUck.