Do I need to transfer my visa stamp to my new passport or can I just carry both?

You can carry both passports if you wish... I have have done it before for nearly 6 months until they put my renewed EOS in my PP... They will never transfer a visa only your permission to stay until stamp 👍

From a legal perspective once the corner of your old passport has been cut, it then is no longer a legal travel document, but you can get your visa transferred to your new passport. Regarding if you mean turning up at immigration to exit the country, they'll want to see your previous entry stamp in your new passport. Now whether a border IO., can or will transfer your visa from old to new is open to speculation. To be safe I'd go to an immigration office and get the visa transferred before trying to leave incase if they let you leave, without transferring it, you may not be able to return on the same visa. It's a good question which I'm sure one of the expat visa posters should be able to clarify.

I went to immigration and they just did it as I came in on the new one but the idiot tried to say I needed a letter from my Embassy until another senior officer told her no and they just did it at Jontiem.