A tourist visa or a visa exempt entry would have given you a 60 day stamp on arrival. You can extend that once at a local immigration office for a further 30 days.
If you’ve already used that option then you can simply leave and re-enter for another 60 days (plus 30).
You don’t need to go to a UK embassy - they don’t do visas for Thailand.
Wrong. Evisa for a tourist visa is $40. Absolutely no point in applying for one as a US passport gets a free visa exempt entry for 60 days anyway.
You are possibly confusing that the visa would be valid for 90 days, but the entry stamp is for 60 days.
There are other visas that are 90 days but they require specific things that a tourist visa wouldn’t. Like being over 50, married to a Thai or having a Thai child.
I currently know nothing about the requirements that may be needed to get an extension. But many have already left the country and re-entered for another 180 days. A holiday to the Philippines may well be your best option.