you will have to search really thoroughly, look over the whole month on a site like skyscanner that allows that but then book direct with airline. My son fly Brisbane to Phuket for around $300 Aussie recently
Go and sell lottery tickets on the street until you are deported after 7 years as just happened to one man. Or go and hide in one of the big caves in the mountains. Seriously though, go and sort it out as if you are stopped and caught by immigration it won't be good.
Avoid these agencies its not necessary. Its easy to get thai license, especially when you bring your current home licenses with you and they will give you thai license with minimum testing. You will need visa, proof of address or long term visa, medical check ssying youre OK to drive.