How should I get married and complete DEERS without spending too much money (Army-Navy inter-service marriage)?

Marriage is tough under the best of circumstances. What you have described isn't anywhere close to "best". Now I'm​ as romantic as the next guy. I just finished the umpteenth cuddle of the day with my wife of forty years to pick up my Droid and read your post. I'd recommend that you see how well you two handle a long distance relationship before tackling a long distance marriage. That will also give you time to explore the options of dual assignments. Trust me, love doesn't conquer all

Search for military join spouse program and contact military one source regarding the regulations for Army and Navy.
Please go through pre-maritial counseling, even if separated to make sure you both reflect on how marriage will forever change your life.

I am excited for both of you! The challenge will be tremendous but my wife and I have been separated geographically many times over overseas deployments and we have been together over 25 years.
Blessings to you both and God Speed!

Without going into the Don't do it, because you are both should be getting that from your respective chains of command. Marriage is expensive whether you start spending money before the ceremony or afterwards. Depending on her rate and your MOS there are not a lot of places for the two of you to be co-located. Get ready for the long distance and if you are actually in it for the right reasons you will figure it out. Marriage is for adults, you make a decision, stand by it, and make shit happen.