Is this variation of TQBF still PSPACE-complete?
It is an Unordered Constraint Satisfaction game and it is PSPACE-complete and it has been proved to be PSPACE-complete only recently;
a proof can be found in: 

Lauri Ahlroth and Pekka Orponen, Unordered Constraint Satisfaction Games.
Lecture Notes in Computer Science Volume 7464, 2012, pp 64-75.

Abstract: We consider two-player constraint satisfaction games on systems of Boolean constraints, in which the players take turns in selecting one of the available variables and setting it to true or false, with the goal of maximising (for Player I) or minimising (for Player II) the number of satisfied constraints. Unlike in standard QBF-type variable assignment games, we impose no order in which the variables are to be played. This makes the game setup more natural, but also more challenging to control. We provide polynomial-time, constant-factor approximation strategies for Player I when the constraints are parity functions or threshold functions with a threshold that is small compared to the arity of the constraints. Also, we prove that the problem of determining if Player I can satisfy all constraints is PSPACE-complete even in this unordered setting, and when the constraints are disjunctions of at most 6 literals (an unordered-game analogue of 6-QBF).

From the content:

...
Our generic example of an unordered constraint satisfaction game is the Game on
Boolean Formulas (GBF). An instance of this game is given by a set of m non-constant
Boolean formulas $C = \{c_1,...,c_m\}$ over a common set of n variables $X = \{x_1,...,x_n\}$.
We refer to the formulas in $C$ as clauses even though we do not in general require them
to be disjunctions.
...
A game on $C$ proceeds so that on each turn the player to move selects one of the
previously nonselected variables and assigns a truth value to it. Player I starts, and
the game ends when all variables have been assigned a value. In the decision version
of GBF, the question is whether Player I has a comprehensive winning strategy, by
which she can make all clauses satisfied no matter what Player II does. In the positive
case we say that the instance is GBF-satisfiable. ..

...
Theorem 4: The problem of deciding GBF-satisfiability of a Boolean formula is PSPACE-complete.

EDIT: Daniel Grier's has found out that the result was also settled by Schaefer in the '70s, see his answer on this page for the reference (and upvote it :-). Schaefer proved that the game is still PSPACE-complete even if restricted to positive CNF formulas (i.e. propositional formulas in conjunctive normal form in which no negated variables occur) with at most 11 variables in each conjunction.