I need some help choosing a GPU (with a thank you giveaway as an incentive)

If you have to pick between the two, no question take the 3060. If you can spend a little more for a 3070 or even a 3060ti that would be a lot better. I personally wouldn't get a 3060 and wait till I could get at least one of the two I've mentioned.

Judging by benchmarks, the 7600 is faster if no RT is involved, but if you turn on RT, the 3060 is faster.
I think Nvidia is more effective with using memory, so the 3060 with 12GB might last longer than a 7600 with 8GB RAM.

TLDR:
Your CPU is old and will impact the low-end (minimum framerates). You will still get big gains from a GPU upgrade.
Both cards are okay for 1080p, not great in all cases, but upgrading from a 960 either card will feel much better.
3060 has more ram, better low-res upscaling (DLSS), and might have an advantage in AI features.
Neither card is well priced in the market. I feel like both cards should be closer to the $200-220 price point.
The RX 6750 XT is around the same pricepoint and about 25% faster. It also uses 50% more power (From 170 to 250 watts)
I usually run my 3060 power limited to 100w instead of 170w and (seem) to only lose 7-10% performance. (Keeps room temp down)
Neither card is fast enough to care about Ray Tracing. Most friends with 3080's wont even turn it on due to performance impact.

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Everyone else provided some great links, so I'll just relate my thoughts and experiences instead.

I had a i5-4670k with a GTX 960 when i upgraded to an RTX 3060. It was noticeably faster, but my CPU definitely held the videocard back somewhat. (Went from 4500 points in passmark on Win7 to 13500 points, and then to 17000 after upgrading my CPU.) Scores were 25% lower than after my CPU upgrade to a Ryzen 7700 on Win10. However, the 3060 didnt solve my 'minimum framerate stutter' issues in many cases, but it did bring my averages up reasonably over 60 in a lot of them. You might still be forced to run on medium with a couple of high settings however based on that CPU. (The good news is that medium usually looks pretty good in most games these days.)

Which you go with depends a little bit on what kinds of games you play, and at what resolution. The 3060 has 2-3 main advantages: 12 gigs of Vram should provide a bit more longevity, but the question is will you be getting reasonable performance or image quality at the point when it gets 'older' and needs to use that Vram. If you want to play a lot of brand new release games, DLSS might give you an overall better experience when it comes to 'running at a lower resolution and upscaling to the native one.' (Ive heard that AMD's FSR is much worse than DLSS, but ive not compared them personally.) Its also said that Nvidia cards do much better with 'AI features' however ive not followed the AMD 7000 series that closely. Not too long ago there were options like the RX 6750 XT 12GB which was quite a bit faster at rendering than the 3060 at the same pricepoint, but lacks DLSS (and AI features i think.) Newegg has them for $300 at the moment in my region.

I have briefly used DLSS at 1080p for a couple of games and barely noticed it was on, but it depends on the games you want to use it with too. Id much rather use native resolution rendering, but if the choices are 'drop below native resolution, use percent based resolution scaling, or turn DLSS on' id usually rather turn DLSS on. Right now graphics cards are in a very weird spot. I think the minimum Vram one should probably go with is around 12, and ideally 16 gigs personally, but that does not give many options for Nvidia cards, and thats hard to achieve at a reasonable pricepoint.

The jump from 1080p to 1440p is something like a 25-35% performance hit in many cases. I do not think the 3060 or 7600 are good value for 1440p 3d gaming unless you want to run DLSS/FSR all the time, if the games you play even support it. I have not had to worry too much about Vram usage with the 3060, its hard to get solid 60fps on any settings that might take the card near 12 gigs of Vram. But i would have to keep an eye on it more often with an 8 gig card i think. Going over the Vram limits can cause textures to just completely go missing, or texture loading stutters in games.

In hindsight i wish i had gotten something a little closer to a 3070 in performance, but that would have been at the cost of ram. I do not think id be happy with 1440p on my 3060 so im limited in my choices for monitor upgrades, i need to stay above 60fps vsync at all times to prevent stuttering (and crippling migraines). If you had a Gsync or Freesync monitor and are only slightly dropping below 60, that might be less of an issue. Right now i think the best 'it does all the things id want and at 1440p card is the 4070 Super, but its nearly impossible to justify $600 bucks for one.

Guys, thank you for the recommendations. I will read everything carefully and respond as soon as possible. My cat needed to undergo surgery, and I didn’t have the time or the peace of mind to keep up with everything here. But now she’s home and doing well! :)