But since the ramming happened at walking pace, people could have evacuated at running pace and still left Shinzon helpless to do anything in time.
Shinzon took out the warp engine with his first shot, so no I don't think firing a phaser at the warp core would've destroyed the ship.
"Warp engine", "warp drive" and "warp core" are different things. When ships do battle, it's quite typical for them to knock the opponent out of warp, probably by destabilizing the opponent's warp field; that means loss of warp drive, at least temporarily. Sometimes the hit may do damage to the engine nacelles, preventing them from creating warp again until repairs are effected; that means loss of warp engine. Only incredibly seldom does anything happen to the antimatter reactor itself, and if something does, the ship usually goes kaboom immediately; that's damage to the warp core, and it didn't happen in ST:NEM.
It's perfectly possible to destroy a warp core with a hand phaser. We have seen such a thing attempted in TNG "Heart of Glory" and VOY "Projections", with supposedly ship-terminating consequences. If Picard really wanted his ship to go kaboom, he could have ordered one of his minions to do that phaser trick, or any of the zillion other things that can be done to blow up an antimatter-laden ship from the inside. But that would have meant condemning those minions to death - and Shinzon, already defeated, simply wasn't worth those deaths.
Timo Saloniemi