If your goal was SWCC then I would have recommended going for that. If you have no medical issues then your recruiter would put you in touch with a NSW recruiter. Someone from NSW will give you the PST and interview you for a bit. If you pass both they'll give you a slot and you ship to boot camp.
You could get a mentor too,
http://www.sealswcc.com/navy-seals-find-mentor.html#.VjKjyPmrSUl
If you want to prepare for it, you're better off preparing while being a civilian. Your job comes first now. But you can still start now and continue your training.
http://www.sealswcc.com/pdf/naval-special-warfare-physical-training-guide.pdf
Basic advice: Take cold showers only. Swim a lot, run a lot. If you make it through, there's some solid advice on the various forums related to this.
But since you're an LS for now, the best advice I can offer is to get promoted and get your college degree. Use the Northstar Study guide. In those 5 years you can get to e-6 no problem.
Also, if you told your recruiter that you wanted to eventually go for SWCC, I'm honestly baffled why they recommended LS. You have no security clearance and the job doesn't help you prepare for SWCC at all.
I would have recommended a job that has a short 'A' school with security clearance. Or something that would provide training that could be expanded upon in SWCC. Something like MM, EM, GM, MA, or HM.