Where you live now doesn't matter. It's all about what passport you have. If your passport is from a visa exempt nation, then you will receive a 60 day stamp if you arrive without a visa.
If the state of your drivers license matches the embassy you chose, then just use it even if it's not your current address. But if the address doesn't match the embassy you are using then I'm not sure how to proceed. Do you have a bank statement with your address on it?
Sounds like you have a round-trip ticket from October to January? If that's the case, then they know you don't have permission to stay in Thailand for that long. You will have permission to stay in Thailand for 60 days when you land if you don't have a passport. The airline may require you to show proof of leaving Thailand within 60 days, as many airlines require this since proof of onward travel in the number of days you will receive on arrival is a legal requirement, extensions do not count.
By the way, Oct 14 until Jan 14 is 93 days, so a visa exempt and extension will not cover your trip. Even a tourist visa will not cover this amount of time, as that is the same 60 days on arrival with a possible 30 day extension. You should buy a ticket out of Thailand and show that you will be leaving the country within 60 days. You could google "onward travel ticket" for services that offer this.
You need to ask your local immigration office what their policy is. Each immigration office sets their own policy on this and it's not consistent across Thailand.
If you're in your home country then your driver's license if it has your address or a utility bill with your name and address on it. If you're in a foreign country then a hotel booking or passport entry stamp.