I'd find a landlord who will Co operate as you'll only be looking for another one in 2 months. Buyers market. My current landlord is great. Last one was a disaster.
Not having your name on a contract could cause you some problems should you wish to apply for the one year extension and not having the tm 30 completed will certainly cause you an issue.
First thing the IO checked at Phang Nga immigration. Sure enough landlord registered me in to the house next door (same landlord) all fixed after 20 min and no fine.
As mentioned there are busses heading north from Phuket. Cheap. About 110thb. A taxi will be between 1200 to 1500. If you take a bus you'll need to get from the airport to the main road (about 3km) Let me know if you need a taxi as I know a very reliable one.
Keep it a simple request from your wife to immigration to authorise a visa for you to remain in Thailand in order to sustain a family life and to appropriately care for your family. Make two copies and you'll probably have to supply the original. Hand written should be OK. Signed and dated.