I think the damage range and potential modifier range is definitely tied to weapon level.
However, when you get into TVHM and beyond, there is usually a flat buff to all enemies--larger HP pool/ % damage absorbance modifier that goes beyond general character and item level for calculating effectiveness in making big red numbers over your targets. BL2 and UVHM was notoriously, universally unpopular for this because it made the slag element mandatory if you hoped to do any damage to anything. The whole point of that stupid element was really to just remove that flat % armor buff for whatever period of time slag debuff was active. It was very stupid.
But yeah, if you are greens and whites, those don't seem to carry through as long. Some blues and certainly purples can last many levels, and some just don't. Some have accuracy problems, or bad recoil, or just weird bullet mechanics that you either like or don't like.
Rule of thumb in these games is to stick to the 2 or 3 weapon classes that you prefer, especially if they sync well with your build, and try out most of what you pick up. fire some rounds at a wall to check out the bullet spread. accuracy is a big deal here, so it could be that the damage is fine, but you aren't hitting what you think you are hitting. Comparing bullet spread on a wall across several guns is a good quick way to check out how well a gun might work for you.
I definitely switch guns out a lot while leveling. Truth is, probably 90% of the guns in these games are just trash. Even the good ones have some trashy stat drops, so you really need to spend some time testing them out, and when you notice that the damage isn't keeping up anymore, you really need to replace it. So far, I'm liking ARs more than I ever have before, and I'm back to liking Jacobs to the quality they were in BL1. Actually, TPS had that really great Jacobs revolver (a couple of them). Nisha was my primary in that, and well just because she was a class, there would have to be some great revolvers.
I was really excited to pick up a purple Masher this morning (reward on the first side mission for Eden-6, or whatever it's called). That was one of my favorite guns from BL1 (TPS had them, sorta...but I think they were called something else), but the damage so far looks pretty terrible. It was right at proper level, but needed 2-3 shots to blow off just about any head, didn't seem to matter how close I was. I need to go fire it at some walls or something to see what kind of spread it is doing, or maybe the damage just isn't as good as it seems? Those things used to blast heads off very effectively in BL1, and with pretty decent medium range. They were like shotguns with handgun range.
--I'm not having too many performance issues, except for some very annoying hitch/CTD. I updated to most current Radeon drivers (I have a Vega 64--the Sapphire 3-fan fancy thing). Most of yesterday it was fine after the driver update, but it crashed again this morning. Otherwise, performance seems "pretty good." I'm 60fps+ @3440 x 1440 using those suggested settings in the HUB video posted earlier...so still DX11. I haven't even tried DX12 yet, so not sure if that still has issues? I saw that there is supposed to be a big update this week....but hey, at least they found the time to "balance" the game ahead of fixing the actual broken things.
And I'm really digging this Samsara build for Amara. I'm not there yet with enough skill points, but the idea is that you have Samsara up near-permanent (only 1 second gap while in a fight), on top of all the ele tree goodies, like Tempest and of course Rush. Not enough points yet, for 50 levels, to get the Rush-based capstone in that tree, but you get half Rush effectiveness, which is pretty good. Samsara seems like the better damage output anyway, and is also tanky.