Ezra's lightsaber's design would have changed the rules of lightsaber combat, but not enough to save the Old Republic. Your idea, however, is potentially enough to save the Old Republic.
There is a major difference between Ezra's lightsaber and your idea. Like Richard stated, Ezra's weapon cannot draw the blade and fire simultaneously.
Ezra's lightsaber was able to significantly influence the tactical circumstances of battle, however. In the finale to Season 1, Kanan borrowed Ezra's lightsaber to fight with the Grand Inquisitor. Kanan had the sense to mix shooting and swordfighting, which was enough to surprise the Grand Inquisitor and grant a measure of tactical advantage. Ezra's reaction to Kanan's strategy was simply one of surprise and "why didn't I think of that?"
Ezra never intended for his lightsaber to do both at the same time. It just didn't come to mind. When he built his lightsaber, what drove his design decisions is the need to not get caught by Imperial spot checks. Therefore, he had the illegal lightsaber doubled as a legal stun pistol to evade detection, and to ensure he is not helpless in battle when revealing his Jedi identity is not an option.
The idea of shooting during swordfighting, however, is a very good one nonetheless (though Ezra can't claim most of the credit):
- As the Clone Wars and the Great Jedi Massacre demonstrated, Jedi are vulnerable to shots they can't block, such as during a saber lock.
- A traditional lightsaber has three Kyber crystals: The main energy focusing crystal, and two secondary crystals that allow the wielder to configure the blade's length and power
- If you can reduce the power output of a blade, theoretically you can also divert part of the power into shooting without sheathing the blade.
- It can be a low powered stun, or maybe even shoot to kill. Who knows?
Basically, it seems the theory is sound that a lightsaber that can swordfight and shoot simultaneously is feasible. That said, the question then comes down to: If a lightsaber's shot is incapable of killing, then would a stun blast be significant against a Sith Lord, a master of pain? A comic I chanced upon a month back showed a pre-Phantom Menace Darth Maul being attacked by a Nightsister with Force Lightning - and the Nightsister implying that only the Sith Lords can endure and resist such an attack.