A new development.
I can split the line at it's vertices. It was converted from points so there are tons of vertices. Now there is no self-intersection and the Intersect Tool finds all the crossover points I want, but....
It also finds the intersections that happen at the vertices where each line was split. This results in tons of points, not just the crossovers.
Though this might help somebody spark an idea.
Also, here is a response from a professor of mine.
"The Intersect tool won�??t find intersection for a polyline itself for sure. If you convert the shapefiles to the Coverage format, you can try the Clean and Build tool under Coverage Tools\Data Management\Topology.
The other possibility is to create a geodatabase with the topology of not allowing line intersect with itself and then try to validate the topology." -Xingong Li (University of Kansas Geography)
I will try the above solutions but please post your ideas!
Thanks!