Good advice, Brandy. A few "pointers"
1. Your child will become eligible for benefits at the time you are named legal guardians - generally, after their biological parents' rights have been terminated. They will NOT cover any services prior to this date.
Depending on the laws of your state, there may be a required waiting period before bparents can relinquish their rights and you can be named guardians. Until that time, fiscal responsibility for the child's care remains with the biological parents.
Make sure when you go to DEERS that you have the child added as a PREADOPTIVE WARD. This is very important as it's a special category of legal guardian. Otherwise, they'll have you fill out the standard 'ward' paperwork which is irrelevant and unnecessary.
Once your adoption is finalized, bring your decree back to DEERS and they'll convert your child's status to 'child' from 'preadoptive ward'.
This might come in handy:
http://www.tricare.osd.mil/policy/fy96/benef57.html
Once the child is in DEERS, you'll be able to receive coverage for their medical visits simply by presenting your ID card. Since you're enrolled in TRICARE Prime Remote, and this is a newborn, you have 30 days from the time of their first eligibility to enroll them as well, duing which they'll receive the same benefits as you. If you don't enroll them by then, they'll be considered TRICARE standard beneficiaries until/unless you enroll them.
While I'd like to say it doesn't matter where you go, it does. Believe it or not, even here in Hampton Roads, VA where there are about 150,000 active duty members there's STILL only one DEERS office that understands eligibility of preadoptive wards and enters it correctly - NAB Little Creek. It's not the biggest base here, either.
Best of luck & congratulations on your connection!
Regina, AMom to Ryan Joshua Thomas