I did this exact thing at HMPO in London last December, as has been said travel with both passports and all is good, just make sure the Passport office realises the importance of returning your old passport swiftly as well. I fast tracked my new passport and had to chase them for my old one as they send them to you separately.
Been married for 7 years however chose to go the retirement extension route, once I get the bank letter in and out of my immigration office (Sakon Nakhon) in 20 minutes including re entry stamp also, if you can afford to have the money in your bank I find this so easy.
Vietnam changed their visa rules yesterday for travellers from European countries, I guess if you’re just transiting en route to Thailand it should be ok, if not check it out on the government website.
Personally speaking my retirement extension is due for renewal in April 2020, I currently go down the 800,000 route, but have just opened a Bangkok Bank account and as from next month will fire 65,000 a month over in readiness for the extension in 2021, want to make sure it’s all joined up, cannot take liberties here!