Drag marks happen when a solid particle hangs up on the edge of the blade or wire and is pushed through the soap. There is no way to avoid this if there are particles in the soap, like ground coffee, pumice, dried herbs or petals, etc. If the solids were only on the top of the soap, you could cut the loaf from side to side to minimize the drag marks, but even then there will be some marks.
As far as the bottom of the soap breaking off, there are two things to look into. One is perhaps you're waiting too long to cut and the soap is getting a bit more brittle as time goes by. You might cut a little sooner when the soap is more pliable. This issue might also be related to the blend of fats you're using -- for example a soap high in lard is usually fairly pliable so it's pretty tolerant of when it is cut, but a soap high in coconut or tallow tends to be more brittle and needs to be cut fairly soon.
Another is perhaps you're pouring the soap when it is very thin. If you are, it may be that the soap becomes a tiny bit more lye heavy on the bottom when it's sitting quietly in the mold in a very liquid state. This issue almost always disappears during cure, but it can make the very bottom a wee bit more brittle than the rest of the loaf.