Are there better solutions to this problem other than adding more human moderators?
yes, automated annoyance of the spammers and their commercial objectives.
First, make sure all past and future links on your site are "nofollow"
see here for full details https://en.wikipedia.org/wiki/Nofollow
but basically it makes your site far less attractive to spammers.
If you can, hide all forum content from the public and only allow logged-in users to see any of it. This is increasing common for forums.
Another option to consider is forcing all links to be just the raw text of the url and not an actual working link. Your users would have to copy and paste manually but at least you might still have users.
Another option is to force all links to go through a url shortener.
Another option is to block all links to domains not on a pre-approved list related to your forum. This might not be practical.
Basically anything you can think of to make links useless to spammers but still usable by your members.
If you can, add "near duplicate post" detection to your forum.
As Mike suggests, IP address is very important to filter against.
It might be worth investing in a commercial geoIP database to help narrow your filter parameters. They are typically about 95% accurate.
Bear in mind that the actual attackers may be "working remotely" and scattered over several countries.
If your target audience is geographically narrow (ie USA only) then you might filter by only letting those IPs have access.
If your target audience can tolerate it, perhaps consider adding considerable random delays (days to hours) to the registration and posting processes. Spam farms are volume businesses and if your site is "too slow" to use they WILL be encouraged to attack an easier target.
Clearly explain why you are doing this so that genuine users are not confused.
Another tool to use is rate-limiting.
ie a new user cannot post for rnd(x) days and then can only post 1 small post per day.
Slowly lessen the restrictions as time passes and a user proves they are not a spammer.
All of the suggestions should really be built-in to your forum software. If not I strongly suggest looking at an alternative.
Forum software that implements all these strategies can greatly cut down moderator manual labour by ranking all suspicious users/posts and letting the moderator focus on the ones most in need of a human decision.
Is this additional traffic going to have negative effects on SEO?
Definitely. But the faster you solve the problem the faster your SEO situation will recover.
Be aware that the google search algorithms will judge your site purely on what they see and not your intentions.
SEO is a complex topic for normal situations never mind forums that have been invaded by spammers. You would be well advised to learn more about it and seek reliable expert advise on your specific situation.