Each province has a "Supply" rating, and supply usage. Generally, 1 human-sized unit uses 1 supply, and if the supply usage is greater than the province supply there is starvation. In the F1 screen you can see the supply usage for each province (SU).
To avoid starvation: Don't move a big army into a province with low supply. The scales (heat and life) affect supply in provinces, and proximity to a fort increases supply. Forts themselves generally have lots of supply. As a Last resort, you can pillage a province to temporarily increase supply (and kill population).
To supply your army: Choose a growth scale, or keep nature mages with the army (they generate 5 supply per nature level) or forge supply items (in the nature path: Bag of Wine, Cauldron of Broth, and Summer Sword).
Cutting off supply lines: Difficult. To do that, you have to control the province of an enemy fortress, choose a death or heat/cold scale and push your dominion into their provinces, kill their nature mages or other supply-providing units (like Vanheim's sacred boars), or isolate their army in a surrounded province, cut off from any castles.
So far, the game is only available Online at Shrapnel, and probably will always be. I bought Space Empires IV and SEIV: Gold from Shrapnel and never had a problem... and I've bought lots of stuff from Amazon... so I'd say you'll have to overcome your aVersion, or forego Dominions II
-Cherry