TL,DR: I change my recommendation to a Canon 7d mark 2. I cannot suggest any 5d for what was requested. I had way too much time to write this evening. Forgive me, I’ve over-done it.
And EF-S lenses do not fit EF full-frame Canons. There was never a crop mode in Canon’s full frame DSLRs, and the lens mount is machined in a way that prevents EF-S lenses from attaching to a full-frame camera because of the reduced mirror clearance. This means you’re balancing the cost of an adapter against the cost of half of their lenses, and I am unable to agree with your cost argument. If they want to replace lenses now, that’s up to them, but that didn’t sound like the plan so far.
As for full frame being less noisy than cropped sensors, I shoot with both a full-frame and crop sensor camera of the same generation, and I spend an unhealthy amount of time comparing them. There are specific conditions we have to talk about for the noise advantage to be true for similar generation sensors: we have to be talking about the same framing and shutter speed (obviously we are), and the full frame camera has to shoot at a larger physical aperture (so, faster than the focal ratio adjusted for the crop factor), so it’s gathering more photons for the same framing, with ISOs adjusted appropriately. In short: you have to accept the narrower depth of field to gain the noise advantage.
If you use that Tamron zoom somewhere within it’s range, and you never stop it down, then yes, it will be less noisy.
If we’ve maxed out the length of a zoom, and then cropped farther in, which happens all the time with smaller birds (the lens is never long enough), or we stop down to reach a specific depth of field, all of that goes out the window.
(The reverse problem happens too. If we’re at base ISO on the crop sensor camera, and have to increase shutter speed or stop down to keep from blowing out the highlights, we’ve hit the other wall and full-frame gains its advantage back again. The problem just isn’t as clear cut.)
In Canon land, the 7d mk 2 was the more popular birding camera for a reason, and is in about the same used price range. Trade offs will always be made between formats. A 5d mk 3 with that 50mm/1.8 would be a great available-light people camera, but would be slower and has an older focusing system than the 7dii. Specifically a 7dii has phase detect autofocus in live view, and a 5diii (as far as I could find) does not, covering one of the original requests for silent shooting.
If we are talking macro, like they’ve suggested they were interested in doing, where we are never shooting wide open, I get even more stubborn. I’m not using hyperbole to say I would never again use a full frame camera for macro unless it was my only option.
I probably shouldn’t be spending this much time writing long-winded rebuttals. 
This isn’t supposed to become a “my format vs your format argument,” I’m trying to match the suggestion to the request. I hope you don’t take my overly long answer personally… I love pixls because it’s largely free of “gear fights.”
@my_dad if you were thinking of upgrading the camera, with the minimum of gear change, I would change my recommendation to the Canon 7d mark II from 2014. It’s still a DSLR instead of a mirrorless, but the build quality and controls are on a different level than your Rebel. It was a very popular nature camera for several years running. It’s about the same price used as the full-frame 5dii that’s been mentioned, but uses all of your lenses with no adapters. It’s also new enough that it includes Canon’s Dual-Pixel phase-detect autofocus in live view, which gives you a good focusing system when using live view for silent shooting.
Like other’s have mentioned, extension tubes are a decent way to get into macro photography. They’re inexpensive, even the ones that include electrical contacts for the lens. The drawback is you can’t focus at any kind of regular distances with them mounted; it’s pretty much macro or nothing until they are removed.
I’d round out the recommendations with a cheap manual flash, if you don’t have one. If you make a home-made light diffuser with cardboard, tape, and white paper, it makes a huge difference for close up photography.
