Heya,
I'll be the nay sayer I guess, and say that it's probably not worth it for you at this time. It's not going to be much different in studio settings, with controlled light and a stopped down aperture lens. Most things will perform very similar and quite well in that kind of condition (a studio setting, with controlled environment). Moving to a 6D only offers you better high ISO advantage, and a slight increase in image quality and dynamic range, but you don't need that in studio application.
The other problem is that you go down to just one usable lens, that 50mm F1.8. Your 18-55 & 55-250 will not fetch much of any kind of resale value either, literally not worth selling, would be ideal just to keep with your 60D. This means you have to spend more to get full frame version lenses if you want to have a focal range back. There are relatively inexpensive zooms that are equivalents to what you have now, but they're not superb. There are some third party zooms that would be a nice change for you though. Note, that 250mm on APS-C is equivalent to a 400mm lens field of view on a full frame sensor. To get that reach back, you'd have to get a 400mm lens. Lenses is where full frame really starts to get expensive and that's a big consideration. There's not much point to getting a higher quality camera (even if it's only a little higher) and then slapping low quality glass on the front of it. Full frame isn't a magic bullet.
Examples (used) of prices of some mid-range equipment that would be good quality:
Tamron 28-75 F2.8 ($250ish)
Canon EF 24-105 F4L ($450ish)
One of those would cover you for a wide to mid-range zoom.
To replace your 55-250 on a full frame, would require, again, a 400mm. So you're looking at a used 100-400L at around $800 best price used.
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My suggestion is to consider swapping out glass before you change cameras just yet. Once you have lenses that you enjoy that can migrate to a full frame, then maybe take that plunge. Just a thought!
Very best,