I don't think rusted cables would stop the top from lifting, but they COULD break. In the high desert, I doubt they'd be rusted either.
I suspect you have a leaking manual lowering valve, which is simply returning the hydraulic fluid to the reservoir as pressure builds. It could also be dirt in the regular lowering valve, which is keeping it open a bit. You COULD, of course have leaking seals in the system, which would give the same symptoms.
If you want to jack it up, I think you'll need FOUR farm jacks and I think you'll have to start them on the ground, getting the top up high enough to place the jacks between the top and the outriggers. You'll need a length of 2x4 to support each side as you remove and replace the jack. Then, I think the farm jacks will have enough lift to get the top the rest of the way up. You can find farm jacks at Harbor Freight.
I don't think you have to remove the cables to do this either.
What I've told you is how I'd approach this problem and I may have it wrong, since I've never done it. Someone with experience will hopefully come on and correct any mistakes I've made.
Hi-Lo 1707T - Tire Minder TPMS on Tow Vehicle and Trailer, 300W Solar Battery Charger, Equal-i-zer WDH
2014 F150 Platinum 4x4 3.5L EcoBoost SCrew