Use an old, but resharpened but to make the first pass, and move slowly, even if it results in burning” the wood.
A resharpened bit is a few thousandths smaller than a new bit. This results in the body being a few thousandths larger than desired. When you make a second pass with a new bit, it will remove the remaining wood. Since this is so very thin, the cut is generally quite smooth.
A benefit, other than a smooth cut, is the new bit will last much longer since it’s removing so very little wood.
One other thing..... make a router table... really it's the best way to do a body, and much safer.
Ron Kirn