I have never heard of anything like that before. The blower motor might be expensive to fix but not difficult. It really depends on what kind of car you have though. I hope you can get it fixed without too much trouble.
I have a similar problem, except my car does not shake. If I am driving up a hill with AC/heat/etc my car loses A TON of power! Normally, I will just turn off the AC/heat until the car has sped up significantly or made it over a hill/mountain.
Get your thermostat checked first. Believe it or not, when that's defective, it can have effects all over the car. The good thing is that it is one of the easiest and least expensive fixes, if it turns out to be that.
We just had the blower motor replaced in our van. A friend of mine did the install but I helped. The blower motor was located under the dash on the passenger side. The motor did cost a bit of change and we couldn't find a used one. But, I would have done anything to get that heater to work. It's really cold out at 6am.
I didn't realize a heater could affect the power of the entire engine. This is good to know. We have one car that is really sluggish in the winter. I may just need to get it checked. It's really getting cold around here.
Okay we finally took the dashboard out to find the blower motor. Guess what it was? Seriously there was a big nest of something (no animals in it) in the blower motor. Once we took it out the motor was fine and the heater was fine! I couldn't believe it.