Attempting to shut off the water to the toilet, however, after turning the knob clockwise all the way, the water still runs. The water stops when I place a plastic bottle under the float ball, however, to replace the components I know I need to shut the water off at the toilet shut-off valve. Should my next step be to shut off the main water shut-off valve into the home?
Make sure the valve stem is turning and not just the handle. Spraying a little WD-40 in there sure can't hurt. If the valve still won't completely shut off, you'll have to find your whole house valve which is usually on an outside wall in the vicinity of your water meter. These gate valves are notorious for failing too so don't be surprised if they don't work too. If that's the case, there is usually another valve on the line side of your water meter that you should be able to operate. If not, call your water company and they will operate it for you. So if you get your water shut off, this is a great time to replace any of these valves that failed to do their job. good luck.