For years I have tried to grow grass in my lawn (a little over a half an acre) but more and more weeds seem to be taking over to the point it literally looks like I have the second worst lawn on my street (my next door neighbor has the worst). I live right outside of Pensacola, FL in zone 8A and I have patches of centipede, St. Augustine, and Bermuda intermingling with all the weeds, as well as bare and brown spots. It is plagued with nutsedge, Florida pusley, crab grass, dandelions, little weeds that look like miniature mimosa trees, thorns, and (now dormant) annual bluegrass (weed type) . I have treated them with Fertilome�s Weed Free Zone, per the advice of a farmer at the feed store, but all to no avail. I also have grubs, fire ants, and mole crickets which I apply Talstar to treat but with my neighbor not caring for her lawn- treating mine only last for a little bit as the bugs cross the line back into my lawn.
I hoped, if I could actually establish my lawn that I would be able to fight the infestations of whatever blows in from the neighbor�s lawn and that leads to my question:
Any way I go- I will need to level at some point before seeding or sprigging zoysia. Should I kill the entire lawn and put down topsoil and sprig/seed
OR
Should I try to save what I have and sprig/seed after leveling?

I am novice at best when it comes to the lawn but I am so frustrated with it looking so bad.
Also any advice on sprigging or seeding, and leveling would be greatly appreciated.