My Time Machine drive is full and is not deleting old backup files (folders).
Do you mean it's failing for lack of room? If so, it's already deleted all but the most recent backup (it won't ever delete your last remaining backup). If so, it sounds like the backup disk just isn't large enough to contain that last backup plus a new one. See #C4 in Time Machine - Troubleshooting for details.
Every article I've seen says that I should go into Finder and click on the gear icon, and from that select the option to delete backup files. However, my options are:
You missed a couple of steps. Go into the Finder, Enter Time Machine and navigate to a backup you want to delete, then click the "gear" icon. See Time Machine - Frequently Asked Question #12.
I know from experience that if I just move them to the Trash folder, that doesn't work properly.
Correct (on Snow Leopard or earlier -- it does work on Lion or later).
There actually are 49 backups in my Time Machine drive. Time Machine isn't deleting the old ones as the disk fills up.
I tried to follow the FAQ #12 instructions, but now Time Machine says that the files are read only. I unlocked the lock icon in the bottom right corner of the Time Machine "get info" box, but it locked itself back. So this is a new problem. Suggestions here?
It sounds like the backups are damaged or corrupted. Did you get this message recently?
If so, see #C13 in Time Machine - Troubleshooting.
If not, what happens when you run a backup -- does it fail? If so, with what message?
Just where are these backups? On a directly-connected (USB or FireWire) external HD, or a Time Capsule, or on your network?
You might try to repair the backups, per #A5 in the above link.