Hi John,
I use the https://moz.com/tools/seo-toolbar to quickly scan the potential and "spamminess" of a link to my site. For a more thorough look you can use the open site explorer (https://moz.com/researchtools/ose). Fact is that bad links are bad and I would definitely try to get rid of them if it is that easy as you mentioned. I also use google Search console to see which links are added for SC is more up to date than MOz. New links you can check again with the toolbar/ open site explorer from MOZ. Since you still rank 8th position on the keyword you haven't been penalized yet so I wouldnt worry too much. Indeed put some (hard) work in high quality backlinks and I am sure you will never regret it 
Good luck!
Tymen