Okay, several things. First of all, there are 2 crankarms, one with the chainrings and the other with just the arm. When you are saying, you spin the crank, you are spinning the whole crank and when it gets to one spot, the resistance gets significant.
Now, as to what you did in the original post. You said you tightened down the bolt. There are 2 bolts or nuts that hold the crankarms to the spindle. The spindle is the axle thing that goes in the middle of the bike through the bottom bracket shell that connects the 2 crankarms. What you said was that you think you over tightened the bolt on the drive side because "they" (not sure exactly what you mean by they) were shaking. I am assuming that what you mean is that the drive side crankarm was loose on the spindle and had play in it however the non-drive side arm did not have any play in the crank arm. So you went and tightened down the bolt that holds the drive side crankarm to the spindle.
If that bolt was loose to begin with, not a good thing. When you tighten down that bolt, you need to use a LOT of torque on it. You NEVER want those bolts loosening up or falling out, if they loosen up , they will damage the crank arm on the square taper. You really can't cause any binding by over tightening those bolts since they aren't bearing loaded bolts. I think was happened was that you didn't notice the binding in the crank assembly until after you tightened down the crankarm bolt but remember, tightening down the crankarm bolt really has nothing whatsoever to do with the bearing tightness/binding of a square taper bottom bracket.
So back to square one. You fixed the original problem of the loose crankarm by tightening down the crankarm bolts, good. However I bet you might not have tightened it down enough, for some reason, people have a tendency to be afraid to tighten down those bolts.
Now you have to figure out what is wrong with the bearings in the bottom bracket. Do you know if you have a loose ball/spindle bottom bracket or a cartridge bottom bracket?