I can think of several ways to do that fairly easily, if I understand your question correctly.
First, why not just use Smart Guides and make sure when you create the last anchor point that it is on the "intersection" of the line you are crossing? That would make them 'touching' and completely eliminate your problem. You can also use the properties panel, when you have a line selected, and simply type in the exact length the line should be (ie., whatever the distance is to the line you want to intersect with).
Perhaps you could also select all of your paths, then shift-click on the intersection line last and use the Align panel to align them to the intersection?
You could also simply draw past all of your lines that are supposed to touch, and then use the pathfinder to chop off the little bits on the end.
Or, you could again draw past all of your lines and then use something like the direct selection tool to marquee drag over all the pieces of your paths that are extruding past the intersection points. Then just delete.