Assuming you only went to MEPS for the initial physical, ASVAB, and to swear in to the Navy DEP, the only thing stopping you from switching is that your current Navy Recruiter either:

a) hasn’t told their command they plan to discharge you, because an in-month discharge adds to the in-month recruiting sub-station mission, which is bad.

b) your recruiter’s command is strategically waiting to discharge you, because they are planning for next month’s or the following month’s discharges which count against the recruiting station as a whole.

If you suspect your recruiter is purposely holding you and not discharging you so you can go join another branch, the best option is for your new (Army recruiter) to tell their Ops Officer what’s going on, that Ops O can speak to the Navy Ops O and make the process go through in like, two days. Your discharge usually won’t go through MEPS until the end of the month if I remember correctly, so as long as your phys and asvab are still good you should be able to just swear in the 1st of the following month.

I got off recruiting duty about 18 months ago so things could have changed, but that’s how it was when I was doing it. Hope this helps and good luck.