Hi - thanks for your feedback!
Re: phase ships - very much agreed; actually working on something that should address that for the next release.
1. The CR as it stands in combat seems to heavily favor the computer, to the point of I start a battle with around 87% and the computer starts at 72%. But I run out of CR WAY before them.
Generally speaking, larger ships have a longer peak time, which won't even count down unless you have matching forces nearby. For example, if all you've got is a frigate, and you're facing a destroyer, it's not going to lose CR at all. It's up to you to bring the fight to it. (If you've got two frigates, the destroyer's peak time will start to tick down, but it's got more peak time to begin with, so wearing it out isn't usually a good idea.)
I'd say if you're running all the way out of CR, you're probably either taking on enemies that are too powerful, or just not being aggressive enough. Stuff like the current phase ships notwithstanding; those are just annoying to kill as they are now.
Regarding carriers, what kind of ship are you flying? If you're in a frigate, I wouldn't expect you to have too much trouble chasing them down. Deploying some faster ships may be worthwhile to help chase them down, in any case.
If anyone has any pointers they can suggest to help mitigate I am all ears. Thanks for reading.
One thing to try is the "Safety Overrides" hullmod. It's available from the start of the game, and gives your ship much improved speed and flux dissipation, at the expense of a much reduced peak time and faster CR loss. The latter sounds like it'd just make the problems you're running into worse, and it might - but it also allows you to play a much faster-paced and aggressive style, and would let you chase down phase ships and carriers more easily, as well as deal damage to other ships more quickly. It may be worth a try just to see how that feels for you.
But, yeah, basically, I think that if you're running out of CR, the main thing is to just consider how you might do damage faster - whether it's more aggressive piloting, using a faster ship/speed-increasing hullmods/skills, etc. Or deploying more ships to help you end the fight more quickly. It's actually cheaper to field several ships at once and end the fight quickly instead of running your CR all the way down and using up a ton of supplies to regenerate all of it.
Hope this helps some!