Yes, according the New Union Treaty ASSRs should be subjects of the USSR, and they also must remain parts of the their SSRs (Russia, Azerbaijan, Uzbekistan, Ukraine) at the same time like modern Russian autonomous districts in Tyumen and Arkhangelsk oblasts. But the NUT remained just a project and wasn't actual after Almaty agreement. Vanishing of "Autonomous" from ASSRs names is just changе of name and didn't mean any change of their constitutional statusI assumed based on what I read that they were, in theory, supposed to elevate them to SSR status (worldstatesmen seems to indicate that the declarations made a change in the formal name to SSR)
there are here some declarations http://soveticus5.narod.ru/85/sborn91.htmI couldn’t actually find the text of any of them to see for myself.
I'm not sure how to correctly designate Tatarstan because just at that time they were trying to join the CIS, but unlike the ChRI, they obeyed the Russian government decisions.