Okay, turns out the downloaded Jellybean 4.3 OS did the trick. For future people who run into this issue:
1. Have Odin 3.07 or higher downloaded (Odds are that you have this due to your phone's current state).
2. Download a copy of the OS you want from here: samsung-updates.com/device/?id=SCH-I535 Go to the dropdown menu on the left-hand side, and select your phone. Don't know/remeber your phone's identifier? Open up the back, remove the battery, and there should be a white sticker with your phone's model at the top. WARNING! Unless you plan to pay, these downloads will take a while. Do this when you can have your computer on for 4+ hours.
3. Once the download is complete, unzip the file, and place it in a location that you can easily find with Odin.
4. Open Odin, and make sure that Auto Reboot and F. Reset time are the only options checked. If there are other options selected, uncheck them and proceed.
5. Click the "PDA" button, and select your copy of the Android OS from where it was stored. Assuming everything has gone correctly, a message should appear in the Odin status box along the lines of "Check MD5... Do not unplug the cable...". Wait for the message "Leave CS..." to appear before proceeding.
6. Once "Leave CS..." has appeared, put your phone into Download mode (Turn it off, then restart by holding down the Vol. Down, Home, and Power keys until you get a screen that has informed you that you are in download mode.). Then, plug your usb cable into your PC/Laptop, and then into the USB outlet on the phone.
7. Once Odin has noted that your phone is attached, you can press "Start", and assuming all steps have been followed correctly, the new OS should download, and your phone will reboot, fully functional. If you are lucky like me, it will even have kept all of your settings!