If you wanted to spend maybe 7 days outside of Singapore, I think I’d do 2 or 3 in Malaysia and then the rest in Thailand. Bangkok is another huge city but it’s also somewhat exotic compared to the other two, and much cheaper as well. Or if you wanted to spend some time on a beach you could instead go to one of Thailand’s islands. Phuket is the easiest to reach, but it’s not very interesting. Kho Phi Phi or Kho Samui are probably better choices because they really feel like islands (Phuket is more of a peninsula) and there is still plenty to do.
Two other suggestions would be India or Sri Lanka. Except for a few beaches in Goa, neither of those attract the party crowd, and both are excellent for scenery and culture. India is cheaper as long as you minimize time in Mumbai and other big cities, but that is good advice for many reasons. Sri Lanka is a bit better organized, and on your budget it is good value and easily affordable. There are good beaches along the south coast of Sri Lanka, but India is probably better for beaches.
If you are referring to days trips and that sort of thing, it will be much cheaper if you book once you get there. In Vietnam in particular it’s amazing how cheap tours can be if you book them through your hotel. I once did a 6-hour bus tour to Mỹ Sơn, which is near Hoi An, and I think I paid about US$6. It even included a boat ride and a basic lunch. That same day tour online would probably cost US$30 because the booking company takes a share and there is very little competition.
In November you can find excellent resort deals all over the Dominican Republic, but Punta Cana is typically the first choice, especially if you are looking for an all-inclusive resort with all the luxury extras at a great price. The resort is actually a collection of little beach areas on the eastern tip of the island, so it's very secluded and well away from some of the confusion closer to the capital city.
Trekking in Nepal during peak season might not sound like the most sensible choice for penny-pinching backpackers. But you can still enjoy the ideal weather without spending a fortune on in-demand hotels and expensive permits if you're happy to swap big-name trails, like Everest Base Camp and the Annapurna Circuit, for quieter treks to Kanchenjunga Base Camp and around the Langtang Valley. Idyllic conditions (dry but not witheringly hot) are also on offer in Rajasthan, one of India’s most vibrant and varied states. Chug between notable, attraction-rich cities like Jaipur, Jodhpur and Udaipur on an atmospheric and wallet-friendly Indian train, snacking on spicy pakoras and sipping sugary chai en route.