if you lower a cards strength to zero with an effect that would kill at strength zero, and the effect lasts till the end of phase, does this make the army of the drowned an instant win if you have seven infuence available.

as the strength lowering effect lasts until the end of phase and the save effect occurs before the army enters the moribund state, thus the effect seems fo stay on the card and therfore it strength would remain zero end therfore would still be under the "Killed if strength equals zero" effect, then this would trigger the "if army of the drowned has already been saved this phase it claims 3 power" effect and would allow the controller of the army of the drowned to trigger this ability repeatedly until he has acheived enough power to obtain victory.

is this logic correct or is there a step in the resolution of the effect on the save that would nullify the death at strength zero portion of the str lowering effect to prevent this from becoming the single greatest army in the game

Sadly no. Though I did manage to build a deck that killed him 5 times in a challenge phase... You would have to have a card like "Risen from the Sea" that actually worked on a character that can't take attachments. Perhaps if your opponent controlled King Balon's Host.

You can find this in the FAQ on the AGOT support page.

(3.19) Terminal Effects

A "terminal effect" is any effect that would cause a card to become Moribund and would still cause that card to be Moribund even after the card was saved. A card cannot be saved from a terminal effect unless that saving effect also removes it from the terminal state.

For example: Flame-Kissed (CORE T103) is played on Selyse Baratheon (CORE B70), who has a STR of 2. Selyse cannot be saved from the effect of Flame-Kissed unless the save also removes her from the terminal state, either by boosting her STR, discarding the Flame-Kissed card, or removing Selyse from the play area. Viserys Targaryen (CORE T108) is an example of a card with a save effect that would remove itself from the terminal effect of Flame-Kissed.

Short version:

As you note saving the Army doesn't actually remove the effect that kills it with a "kill at 0" effect. According to the FAQ, it is illegal to use a "save" effect on a continuous kill like that unless it also removes one or more conditions of the continuous kill. So the Army's save effect cannot be used against burn effects.