A few suggestions:
Upgrade your companion gear more often. I actually find myself spending as many badges on my companion's gear as my own (since I get upgrades like crazy and she does not). I also can craft some armor for her, and as I level up Armormech I make sure to give her upgrades I can make. I don't know your profession, but if you can earn crafting skill AND up your companion, do so. If not, seriously consider investing a few badges in your companion's 15 level old gear before upgrading your 3 level old gear.
Make better use of CC, stuns, interrupts, and openers. Interrupts, especially, if you have any. It can be a bit tough to notice an enemy cast with all the other stuff going on, but it can make a huge difference. As it's been said, stims and buffs are always helpful as well.
Have you tried a healing companion? Everyone can use their ship droid for this purpose. He's freaking annoying, so I wouldn't keep him out all the time. But get him some parts and keep him in your pocket for those one or two fights where you just can't survive long enough. Having a pocket healer on a tough quest is awesome, and I'm pretty sure that's why they built the ship droid that way.