Slitherine

So I've just been messing around with the different scenarios, and I was curious to what I should do from the start of the 1944 scenario as the Axis.

In case some don't know the situation,

Italy: Out of the war, American's in Rome, German units control northern Italy.
France: D-Day is about to start.
Eastern Front: on the verge of total collapse.

What do I do to prolong the war, and if possible even win, vs. AI or Human player?

Get down on your knees and pray.

But seriously, I've looked at this scenario but never played it. If memory serves, the eastern front is non-linear, with a major bulge in the center. Your first priority should be finding some way to straighten this front, thereby shortening it. That probably involves a limited withdrawal in the more exposed areas. Also, shift all your air forces to the east. The Western Allies dominate the air in the west; don't try to challenge them. The Luftwaffe might be able to do some good against the Soviets. The British and Americans will have a harder time getting to Berlin if you don't stop them than the Russians will. Plan a defense at the Seigfreid line and in Holland. Get all your stronger units out of Italy, maybe just leaving a few corps to delay the Allies. You need all your tanks and mechs to slow the Soviets. Your oil is probably low too, huh? You'll have to be careful not to run out.

I think this is the strategy I would start with. If you play a human, you will probably have to devote more resources to delaying the Allies in France than you would against the AI. It's probably only possible to win if by "win" you mean last longer than the Germans did historically.

1944 scenarion is quite easy for axis, compared to situation what happens in regular game by this time.
You have lots of good german troops with no manpower penalty, you have some oil, you have lots of free commanders.
Just dont forget to sentry your fighters as soon as you can, or they burn all your oil.
You have to hold on for just 20 turns, half of which will be bad weather, nothing hard, especially vs AI.
Try to pick good defencive positions in bad terrain/behind rivers, entrench and fight. Destroy any breakthrough with armour reserves, you have lots. But don't leave then exposed after attack - shiled with infantry.

Its very easy to score minor victory via holding Berlin, a bit more hard to hold Paris for major one, challenging to retake Rome for strategic and almost impossible to grab something 4th for ultimate.