We're in the middle of a(n epic) flood. Why won't my toilet flush?
Scott,
I'm not a plumber but I think I have been through just about anything plumbing issues can throw at you. I think you provided enough information to get it figured out. I don't think it's item one below since you said it's only an issue when you have torrential rainfall. BUT. One comment made me address this too.


When you said the toilet fills up somewhat when flushed, it leads me to believe your tank is not filling up and getting enough water for a complete flush. - Pull the lid off the back of the toilet and make sure it's filled up before you flush. (should be a water mark / stain showing where the normal level is) If it's not full, it won't flush properly. Get some water from another faucet and fill the tank up and flush. If that works. Good. If not, you have another issue.
Since you are in a municipality (mud district), your house and the neighbor directly behind you are piped into the main sewer line. (All lines run out the back at connect to a center trunk basically. There should be a PVC cap just behind your house (maybe 2-3 feet from the house). This cap should be screwed on (threaded - not glued) This is there to remove for cleaning the line out going either way. (toward the center trunk or under the house and into the toilet). Take that cap off and see if it's full of water/sewage. If it is, your line is either plugged up, collapsed or the main sewer is full.
Good Luck