I used to shoot with a Canon 7D, which was a fantastic bit of kit but I had a little FF itch that needed sorting, similarly to yourself. I took the plunge on a 5Dmkiii about 2 years ago. It was head and shoulders above my 7D in every way you'd expect a full frame to perform over a crop sensor...
but...
I loved the 5Dmkiii but wanted a more tactile shooting experience, with a camera that I enjoyed physically using to take photographs. In this respect, I found the majority of Canikons to be a little souless. I guess ultimately a Leica would have been the end goal but...money. I bought a Fujifilm X-T1 to shoot alongside my 5Dmkiii and loved it, but it was too slow on autofocus and wasn't quite "there". Then I made the upgrade to an X-Pro 2 a little over a year ago and loved it. I currently shoot 3 X-Pro 2s for work, and have an X100F for fun.
Ultimately, I suppose I'm maybe saying it depends on what you want from both your pictures and your shooting experience. If finances weren't a consideration then picture quality and low light performance hands down would be in favour of a 5D mark iii or as others have suggest, perhaps the mark iv would be a wiser investment. Build quality on these cameras is fantastic and again kicks the **** out of my Fujifilm cameras. It looks like you've got some good glass so perhaps switching systems isn't something you'd consider. As a camera I'd take literally anywhere with me however, and for the tactile shooting experience I can't fault my X100F, which is by comparison very small, but you're
very limited for focal length options (it's a fixed lens camera) - I would however encourage the idea that sometimes fewer options can be somewhat liberating. Not suggesting necessarily that camera specifically, but if you're thinking of making changes then perhaps consider other options too. I know that I'm waffling away from your question a little, but something to think about.
If I flat-out had to choose from the options you've been considering I'd get the 5D mark iii.
If I had to recommend something loosely based on what you're considering and budget wasn't a factor, I'd be another for the 5D mark iv.
My final thoughts on your predicament would be to rent the camera body you're seriously considering.
Would love to know the end result of your decisions!