i just aquired an old Ross Signature with 700x28c tires (also listed as 28 x 1 5/8 x 1 1/8 on the tires).
the bike is in seriously amazing condition, except the tires have a ton of wear and are beginning to split.
i've been reading and rereading Sheldon Brown's articles on tire sizing, and my question is, what widths of tires can i put on my bike?
i don't know the size of the rim, and i don't have any way to measure it, unfortunately.
if i put a 700x23c tire on, would it just come loose?

Can't tell from the info you've given. But I would guess that a bike with 700x28C tires would have 17mm (inside width) rims. You can go as narrow as 25mm on those. I once tried putting a 23mm tire on a 17mm rim and the tire bead would not seat properly and the tire blew off the rim when I pumped it up.

Depends on the type of bike this is. 700x23 is pretty narrow, usually suited to fast road bikes. If this bike has 700x28 tires, it sounds more like a touring road bike or hybrid. It also depends on your needs. Do you want to go to as narrow a tire as possible? If so, you could probably go down to a 700x25 and the bead will still catch the rim. But unless you want the reduced rolling resistance and added speed of that narrow a tire, I'd stick with one as close to the original as possible.

Get a pair of Rivendell RolyPoly 700x28. I have those on my bike and they are the best cornering tires I have ever ridden. https://www.rivbike.com/

Most sport bikes can atke 25-32mm tyres.
There is wide variation in what exactly a 28mm tyre is. Some are light and narrow, others are wide and tough.
If you pick a narrow brand (like Panaracer), then the 28 will fit, and be quite light/fast, but their 25mm may be on the small side for your rims. The only way to tell the real size is to compare it to your current width, by sight.

Go Back to Sheldon Brown's WEB site. There's a table there that gives the acceptable tire widths as a function of inside rim width. 700X28 fit fine on a 15 mm rim (my wife has that combination) which is a very common road-bike width.

Al