Why does the blower in the attic vibrate during heating when it is really cold?
Have you tried placing a clean fresh filter? Probably, but I have to ask. The real problem is the blower isn't well balanced and other issues exacerbate it. I've not heard of anyone balancing blowers, but I don't know everything.

Another possible issue is the blower anchorage is loose. Some are mounted through resilient mounts. The rubber may have perished, fresh mounts could help solve the problem.

An under sized duct system could be a contributing factor, addressing that would be quite expensive. Be sure all your registers are fully open. There may also be balancing dampers in the ducts, you shouldn't mess with those unless they look almost fully shut. Or it could be the blower is over sized and you do not need that much volume.

Sometimes the speed can be adjusted. If it's truly over sized, you may get by with only medium speed. Depending on your system, an inappropriately low fan speed can be dangerous. Be sure a qualified technician is making the decision to lower the speed.