The number you gave us is the number inside the tank, which isn't the full model number for the toilet (so I don't know for sure the color and gallonage used per flush), but it does tell us a lot. The lid for your Carlyle is going to be part number TCU854. If it is the 1.6gpf version, then the full lid part number, if it is Cotton White, is TCU854CRS#01. There are a number of online channels selling used lids, including the famous auction site. I would start with your local plumbing supply to see if they have or can order the part, and how much they would want for it. Make sure you are getting the right color.
You can also get it directly from Toto in their online store. Go here: Scroll down to the TCU854 listings. You don't want the bolt-down lid and you need to make sure you get the right color. TCU854CRS#01 is shown as available for $69. Other colors have a different number after the # symbol, and different prices. The phone number for Toto in Georgia is 1-888-295-8134. They have friendly people who can help you identify and make sure you are getting exactly the right lid, and they ship quickly. It may be cheaper through the secondary channels, though.
I can say that I did purchase a lid for a 50-year-old toilet online not too long ago, and it came intact and well-packed. There was a big crack in the old lid and I otherwise would have had to replace the old toilet just because it was so unsightly. New lid, new handle, new Korky guts, and I was back in business and looking good for about $60. (This for a toilet that gets very little use.)
You might also ask about the toilet seat bolts, but you are probably out of luck with the manufacturer. A good hardware store might have a suggestion. As to seats sliding around, I can say that I like the mounting system on the new Mayfair slow-close seats with the lift-off hinge feature. (e.g. model 148E2) You actually mount the nylon bolts on the toilet without the seat attached, leaving a big nylon head above the hole; the bolts and mounting system makes them stay put. Then, you clip the seat to the bolts. On our toilets where they used to slide all the time, they don't move at all. You want the absolutely-newest version of the mounting stuff, as it has changed a little in the last year, and the most recent incarnation is the best. You would get that by buying online or locally from a company that has substantial inventory turnover. Unfortunately, the seat is a hard wood seat, which we like but I know many people want the more-malleable plastic because they find it more comfortable.
Also, were you able to solve the "won't stop running" problem? That usually just involves replacing the flapper (the rubber thing on the flush valve that holds the water in the tank). Assuming that you have the 1.6gpf version with the red or pink flapper, your Carlyle will take this one as the replacement: If you have the blue flapper or a different color than red or pink, let us know and we'll aim you at the replacement.