Can I apply for a 60-day tourist visa for Thailand while in Thailand?

Cannot. You have to do it while in Vietnam, at the local Thai embassy.

Any news on the Thai visa exempt yet 60 days??

Absolutely no

No.

Start out 60 days in Thailand, and then go to Vietnam, and come back to Thailand visa excempt.

Or start out by going to Vietnam, then go Thailand.

With tourist visa you can stay up to 60 + 30 days with touristvisa applyed for from home, the visa is valid for entering Thailand up to 90 days from the date it is granted.

You cannot apply for a tourist visa while you are already in Thailand.

If you want to apply for one in SE Asia, then you would need to look at the website of the Thai embassy or consulate you want to use and see if they require an appointment. You would also look at their requirements as well as their processing time (how many days it will take).

If it requires an appointment then you would need to book one. And then make sure you can meet the requirements. Some of them require things like a fully paid hotel booking for your entire stay. And all of them will require proof of a ticket out of Thailand within 60 days.