There are many groups on Facebook for gig promotion. You can join & promote your gig. But it can also attract spammers.
So, I think the best solution is marketing through social media pages. If you are not comfortable promoting your gigs through a personal Facebook profile then You can create your brand pages on multiple social media platforms to showcase your work with your Fiverr profile link & attract more clients. Instagram & LinkedIn are the best platforms to showcase your work & promoting your gigs.