The first thing I would try is to turn on the heat and set the thermostat high enough to run and see if the problem presents itself in the heating cycle as well, this will narrow the problem down. If it runs ok in heat then chances are good you have a problem with the furnace control board's fan relay contacts which activate in cooling and fan only. The only option here is to replace the board.
If the problem presents itself in the heating cycle then the problem is more likely with the fan motor or capacitor. If the blower motor run capacitor is weak, it will behave like this and needs to be replaced. These are very cheap so it's worth it to summarily replace it to see if that helps. I should caution you, capacitors (especially blown ones) can hold a nasty charge even after the power is shut off to them. So with power shut off, you should take a screwdriver, and only holding the handle, short each capacitor terminal to the unit casing with the shaft of the screwdriver before touching them. This will ensure the capacitor is discharged and won't bite you!
Tight bearings in the blower motor or a problem with the blower wheel can cause this problem too. You should check to see that the blower wheel turns freely without rubbing and the motor turns and coasts easily. If not, chances are the motor bearings are worn or the blower wheel is rubbing on the housing. You should hear the blower wheel if it's rubbing.
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