What we know, for certain, is that without the dilithium, the TOS ship "loses power."
Perhaps, perhaps not. When three dilithium crystals in a row were burned in "Mudd's Women", Spock suggested "rigging a bypass circuit" so that the one remaining crystal wouldn't be a risk factor. Scotty couldn't oblige because "the whole converter assembly" was damaged. The implication might be that the ship could run at warp or otherwise traverse interstellar distances without dilithium, if only the converter assembly (whatever that is) were in working condition and some special arrangements were made.
In "Alternative Factor", all four active crystals (nice continuity with "Mudd's Women"!) were damaged by the hiccuping universe, and were brought to some sort of a nurturing station. They were then stolen from there, two by two. At no point was the ship said to be deprived of power. Either Scotty managed to rig a bypass circuit this time, or then there were additional crystals that could be taken from storage and put in place of the ailing ones.
Just to nitpick. But it pays to be careful.