You're a UK resident, so you can enter on a visa exemption, which since Oct. 1, will give you 45 days, with the possibility to extend for another 30 days inside the country. That will give you 75 days. You can leave the country and re-enter to get a new stamp. If you need 83 days, getting the Single-Entry Tourist visa would be better, since that would give you 60 days, and possibility to extend for another 30 days inside the country, for 1,900฿.
According to the Thai eVisa website, as a UK you can apply online.
Like Tod said, you can get as much as 9 months out of a METV, however, you need to enter Thailand as soon as you receive it. Weeks not having entered is weeks lost at the end of the expiration date, where you can do the last re-entry.
It would be good if you said if you've gotten a Single-Entry Tourist visa, obtained at an embassy, or if you got a visa-exemption stamp, and how many you've gotten so far. Your post only mentions "tourist visa," which means a SETV/TR.
You're allowed two visa-exemption stamps per calendar year by land. Extension is an extra 30 days, WITHOUT having to leave the country. Doing a border run for a new VE gives you 45 days, with the possibility for a 30-day extension. Why wouldn't you get the extension? It's 40% of the visa.
Flying out and getting a tourist visa at an embassy requires a bunch of time, money and effort. You might not even get it (good chance you will if your passport isn't filled with visas/stamps). And it'll give you 60 days, with possibility to extend another 30 days. You also have to pay 1,900฿ twice, and can't get many.
An option would be to leave the country to get the Non-Immigrant ED visa at an embassy, but I guess the documents would be different than the documents to switch in-country?
Either way, all this is mixed up. OP applied for studying at a school very late.
I think Brandon meant about leaving the country before getting the MOE letter and documents, since he says leaving the country during the process could create issues.
However, even if OP gets the documents in time, he got his extension Tuesday, so he wouldn't have the needed 15 or 21 remaining on his extension to do the switch to the ED visa. Difficult to say if getting a new VE would or wouldn't create issues (different visa, entry stamp, and TM30 than the documents).
Yeah, but I meant, even if he were to have all the ED documents ready in less than two weeks, he would still need 15 or 21 days remaining on his extension to switch to the ED visa. OP said he did his VE extension last Tuesday. So Brandon's suggestion that he still has a month left is incorrect/misleading.