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When you run Spatial Join, you can specify a Merge Rule - Count on a field. In your case, for each line you can get a count of matched dam features (FIDs).
Thank you for letting me know. You are welcome. I am glad it worked for you. 🙂
I think Featue To Line tool (ArcInfo license) breaks lines where they intersect or share a segment. Each output line carries the FIDs of all the sharing lines, or -1 for no sharing line with an input fc. You would need to add a field to the result and calculate the count of FIDs that are not -1 by an expression.