If you have the 60 day visa then go to immigration and extend it.
If you don't have a visa and you extended your 30 days visa exempt entry another 30 already you will go into overstay. Can pay on departure but don't get caught by police before you get to airport as they can arrest you.
Can try apply for another extension and hopefully get the extension declined 7 day stamp to leave
If someone says not required and they go without it and the IO on the day asks for it then you have wasted a day hence why I suggested taking it just incase
You need passport, TM30 from where your staying (some Immigration dont ask for TM30 but take to be safe) copy of arrival page stamp copy of data page 1900baht cash. TM7 form available there to complete can be found online but again some prefer their own form completed, I downloaded for extension few years ago and Bangkok wanted theirs so sat at the IO desk and did their form. There is an office to do coping there if you can not do elsewhere.
Airside means your staying on the international side of immigration and customs. When you transit say throught Singapore like I do every 2 weeks I stay international airside and just go to my gate for next flight to Thailand. Same when I was flying perth bkk to Phuket stayed international side in bkk and got my arrival stamp in Phuket from immigration