When you mention, “not too touristy”, the first thing that comes to mind is San Juan, Puerto Rico. I’m not sure if you’ve been there already or not, but one great thing about it is that the “tourist” areas close to Old San Juan are filled more with apartments than with hotels, unlike Cancun, where it’s nothing but hotels and timeshares. I can highly recommend the Condado neighborhood, which does have some large hotels but is mostly apartments for expats and seasonal visitors. There is even a free bus that goes from there to the heart of Old San Juan, which is a gorgeous colonial town filled with great restaurants, bars, and interesting attractions.
Southeast Asia is easier to deal with, and there are cheap flights connecting in the Middle East and landing in Bangkok, Kuala Lumpur, and Singapore. Each of those is a very interesting city, and there are many smaller places to go in the same region. In Malaysia there is Penang and Melaka. In Thailand there is Chiang Mai and quite a few others in that area. In Laos you can visit Luang Prabang, which sounds like it could be a winner for you. And there is also Siem Reap, which is home to the amazing Angkor Wat temple complex.