You’ve been to Cancun, but have you been to Playa del Carmen? Personally I much prefer it because PdC is a lovely tourist town with hundreds of little hotels and restaurants and bars, unlike Cancun, which is mainly a long string of high-rise beach hotels that are so spread out that you are nearly trapped in your own hotel. It’s about an hour south of Cancun airport by taxi or shuttle. That would be another of your cheaper options, partly because flights into Cancun are pretty cheap. It will be the tail-end of the rainy season in all of these places, but that usually just means a 30-minute rain storm a few afternoons each week.
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.