Is there a cooldown period for reapplying for a Multiple Entry Tourist Visa in Thailand?

Although I do not know of a specific rule, I would bet that there is indeed a de facto cooldown, if you used the visa for near-permanent stay. Thailand's embassies/immigrations appear not to be very keen to let people stay for a full year on tourism-purposed visas or alternatives. Things get usually more difficult past the 6 months mark. So they may weigh how you used the visa in deciding whether you qualify for a new one. If you travel to Thailand very often, but not very long, and needed the visa e.g. to be able to enter by land more often than twice, then I'd expect them to be ok letting you continue that. If you pushed your stay on the former visa to the max until almost 9 months, I'd expect them to not give you another one immediately after.

If the visa you were issued has a final "use by" date in March, then just leave and return right before your visa expires and get a final 60+30 days. That should get you until June.

When you get another one is irrelevant because you can only get a multiple entry tourist visa from your home country, so if you want another you'd have to return home anyways. And it makes no sense to get another one when you only want a few more months.

Either go to a Thai embassy in a neighboring country and apply for a single entry tourist visa, or just do 2 land border bounces.