Dual military TDY to AIT; Will I still receive my BAH at single rate since I will be living in the barracks?

Realistically (assuming you have no children or other dependents), your husband will be able to keep his current lease or obtain housing and receive BAH and when you arrive to your duty station and move in you would begin receiving partial BAH (assuming your husband outranks you?) While you are in training you are considered single for housing purposes and if staying in the barracks you would not be entitled to BAH.

Make sure to update your DA 5960 anytime you gain or lose dependents, move, PCS, or change marital status!

You shouldn't be receiving BAH at the single rate now, and your BAH will continue as it is while you're TDY enroute.