It sounds to me like you are running your piece between the fence and the bit. This should never be done. The bit should be in a space in the fence and only the needed portion of the bit sticking through the fence.
End routing is a bit tricky, since end grain likes to grab the bit, and the bit will move the piece of wood out of square when it does this. If using a miter gauge during the end cuts you need to clamp your piece to the miter gauge, so the bit can't pull it out of position. A second piece of wood behind the piece that you are cutting, positioned so that the bit cuts into this second piece as it leaves the first piece, will prevent the end of the cut on your good piece from splintering as this second piece will hold the trailing edge of your good piece as the bit cuts it, and the splintering will occur in the second or scrap piece as the bit exits from that one. Better than a miter gauge, is a coping sled. It has clamps for both the good piece and the scrap piece, and handles positioned safely away from where the cutting takes place. There seems to always be something else expensive to buy when woodworking, doesn't there, but then when you don't have the money to buy the fancy one, you can study it, and then make your own out of wood scraps. My first coping sled was shop made, and it lasted for years.
Central North Carolina