Has the Army ended the GEO BACH program? Do all dual military members CONUS collect BAH?

As you stated, dual-military have one servicemember at BAH without-dependents rate (you) and the other (if you have one or more dependents) at the BAH With-dependent rate (her).

When the two of you are maintaining separate households due to military orders, your authorizations are considered separately*. In other words, you are considered as a single E-5.

Looking at Fort Liberty’s policies*, this guidance is also reflected there – “Bona fide single Soldiers in the ranks of private (E-1) through sergeant (E-5) are required to live in the barracks … Unaccompanied Soldiers married to another Service member on separate tours will be assigned to housing on the same basis as unmarried personnel.”
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* DoD 7000.14-R Financial Management Regulation, Vol 7A, Chap 26, Housing Allowance - https://comptroller.defense.gov/Portals/45/documents/fmr/current/07a/07a_26.pdf
* Fort Liberty policies - https://home.army.mil/liberty/index.php/about/garrison/directorate-public-works/dpw-ho/unaccompanied-housing

Unfortunately, finance is correct. Since you are different bases and you are not a SSG you have to live in the barracks.
I tried to submit an Exception To Policy to this for one of my Soldiers awhile ago saying SM needed a residence family could stay at while visiting but it was denied. Maybe having a child will be enough to get it approved. I suggest talking to housing about an ETP format. The ETP will have to be approved by all commanders all the way up to Garrison to get BAH. It will take time and be frustrating but good luck to you.

My apologies for the double tap, I failed to add that maintains a household of any size to support your family is way too costly and would likely be a burden on your Family which can lead to problems in the unit. I have never seen it work out in the way that the Soldier intended.