And OP, ask them to help you find long term positions. It sounds like your resume shows a lack of commitment and it'd be good to show you can stick with something for more than a year.
Yeah, a headhunter might be a good idea. I'll check out some which were mentioned here. I heard of triplebyte but have heard they've done some sketchy stuff (spam emails, exposing PI) etc. But it sounds like they do have incentives to champion applicants.