Is it possible to switch from the Army to the Navy and try to get a SWCC contract?

I don't know about what the Navy will offer you when you enlist. I can tell you that there is no such thing as a conditional release for AD Army in this situation. Other branch recruiters will tell you there is (AF recruiters love telling Army this) but you will spend months spinning your wheels and fretting without ever getting one approved. If your command is unknowledgeable enough to sign it, it's still worthless because the only ways to separate as an enlisted person are listed in AR 635-200. Meaning when that recruiter goes to access you into a different branch, you'll still be in the Army.

Sure it's possible, but it's not likely. How long is this first enlistment of yours? Generally when people do get a conditional release to join another branch it's usually as a commission in that new branch. Essentially what you're saying to the Army is I want to go live at mom's house.