I renewed my UK passport earlier this year. I live in Khon Kaen and used an agent called Thai Visa Express, soi 15 Pattaya.
For 10,000 baht they did all the work including colour copying the old passport, going to and from Bangkok, paying by their credit card etc etc. I only had to fill in the application, which they also provide, and supply the photos. If you have not changed much in the past ten years the photos do not need counter-signing. I am not associated with them in any way but as a satisfied customer I would highly recommend them.
You should apply immediately after doing your next 90 day report which gives you 90 days before needing your passport again. But I'm sure they would return it if needed. But they need it back to collect the new one.