If you have a sha512crypt hash and you know it's the same password, then yes you'd absolutely target the sha512crypt hash instead.
I'm not sure which online services/lists you're referring to, but I wouldn't go that route unless you know the site is reputable. Even still, sha512crypt is still pretty damn slow, so you still need highly-targeted wordlists.
I'm not sure which online services/lists you're referring to, but I wouldn't go that route unless you know the site is reputable. Even still, sha512crypt is still pretty damn slow, so you still need highly-targeted wordlists.