As you said in the last line,
"but they often carry ridiculous legendaries or fight in a group"
That is what you need.
First, fighting with a team is very important (if you do not have the Legendaries, and even with them). More people means more bullet sponges. And, with multiple characters, you can fill some roles left open by just one person.
The Legendaries also are very important. The "gimmicks" they give can really turn the tide in a battle.
But, assuming that A - you have no one to play with, and B - have terrible luck with Legendaries, I can give some helpful tips for UVHM.
First, use Slag a lot.
Increased duration of slag damage multiplier effect to 8 seconds.
Increased the damage that slagged enemies take from non-slag attacks from 2x to 3x.
If you Slag enemies, then attack them, the damage is increased much more than in the earlier modes. To help complement that even more,
Weapon swap speed increased to better facilitate slag use.
You should use your Slag pistol until the enemy is slagged, then swap to your highest DPS weapon. But, I would recommend a Slag Rocket Launcher, or (if you have Tiny Tina AoDK) getting the Magic Missile instead of a pistol (you can Area of Effect lots of enemies with Slag).
Now, for your Tree, I would go with this one (I use it).
The Intrepid Outburst allows you to get more Slag on the field, and the bonus shock damage will help with your DPS. But, you would have to go get one or two shock weapons. All of the extra Shock damage given by that tree plus the Slag will do lots of damage. Also, that build is more tanky than the other one.
Lastly, you will need a good shield. I mention the Bee later, but it does not work with Deathtrap (the Amp part at least) (the damage on your bullets though is outstanding). I would recommend either a good Anshin/Pangolin shield, or if you want to farm a boss, the Sham (BNK-3R). They would help with survivability.
Once you get over leveled a bit, you should go farm for Tourge Tokens(on low difficulty). Once you get a lot (around 600, don't remember exactly), keep saving and reloading at a Tourge Token Vending Machine until a weapon called the Double Penetrating Unkempt Harold appears. Buy it, and use it for a weapon ( I reccomend this becuase it is pretty easy to do, and uses less luck than farming for one via killing bosses or enemies).
The DPS for it, along with the Magic Missile (preferably purple variant) can easily destroy enemies.
Getting the Bee shield can also help (no DLC needed), but you should try to get a more tanky shield. A really good weapon is the Florintine (Shock + Slag) becuase it works with the Shock tree, and it Slags (but you need Seraph Crystals).
A good weapon that synergizes with the Bee is the Sandhawk. All you need is the Pirate DLC, and to go through the story. It is a mission reward, so it does not require luck (unless you want to get a specific variant).
If you are playing on Steam, you could try and check the Steam forums for some other players who are looking for groups (who are same level). That could get some more players who are willing to play with you (and you might make some new friends). You could also find someone who is willing to trade/give Legendaries (but many consider it "cheating").
NOTE: This is an incomplete answer, and I plan on editing it more when I get back to it. I must shut down this computer, so I will post it for now.