Do I need to leave Thailand to apply for a new TV visa after my EVisa expires?

You can also buy a one-time per entry 30 day extension at your local immigration office for 1900 baht

Yes, you have to leave Thailand at some point(s).

The date important is now what is the date on your passport, not on your eVisa PDF anymore.

So, if the "until date" on your entry stamp is the middle of February, you cannot continue to stay until the middle of May as a tourist.

If you want to get a new visa, you need to apply for it at the Thai embassy/consulate in another country.

If you mean you want to enter on visa-exempt entry for 30 days, you can do border-bounce (leaving and entering Thailand) for visa-exempt entry. Be careful with your schedule. The same-day border bounce may or may not cover your plan until the middle of May.