Abuse@namecheap.com is indeed the best channel. Our Twitter reps are quite responsive too. The abuse team reviews every case. In some cases, it can also be effective to reach out to the "host" too.
There is a backlog. Like I said, they're a very busy team and are quite thorough in their review. Feel free to ping @namecheap on Twitter with the ticket number and they may be able to help escalate, depending on the severity of the case.
And do you have a better way of going about flagging these sites other than the “abuse” email that shows when doing a WhoIs lookup?