The A.I. does cheat in resources on the "Unfair" and higher settings. I don't exactly know the amounts of extra resources/free technologies etc. that they actually get, but it is quite noticably higher than you can match by controlling the same planets they have. This is to make them more challenging, and as much as I'd prefer to see the game's A.I. win by use of smarter tactics and better playing, it is the way that the devs chose to make harder fights. As I'm still figuring out the game again, I have the same "draw" problem as you when playing on very high difficulty I sometimes get locked into fight that I can't win, but am not losing, usually over one or two key planets. this is generally a good time to invest in superweapons.
Galactic Civilisations had the same technique to provide greater challenge, as do a great many strategy games. It is annoying, I have always thought the A.I. should have to operate under the same constraints as you, but it is really common. It's a pity though- GalCiv had a nice smart A.I. at the higher levels, it didn't need to cheat just to stay out of fights it couldn't win, which is similar in a lot of strategy-genre games (some A.I. in games suck.In very old titles, sometimes there was no way to make the computer challenging besides giving it an army twice as big as yours) but now A.I. programming is good enough that challenging opponents don't need to cheat.
I don't really know any good tactics for getting through this problem, but starbases and very careful use of frigates and cruisers to make a decent counterattack both help. having a titan to wait for your enemy to attack you and level while it joins your defenses in wiping them out can also give your biggest ships an edge. Other than that the early expansion phase is key, you need resources more than normal so colonise everything with an ore-bearing rock over it.