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The way to avoid the wet season is to pick your place; for example in Malaysia you need to be on the west coast until about May and then can move to the east coast as most of the monsoon would be over.
China is HUGE so you'll need to do some research on weather there!
Expense wise, Japan is expensive, then China. I found Malaysia cheaper than Thailand and hear that Vietnam and Cambodia are cheaper still. For example, there's an excellent hostel in Malacca, Malaysia, where you get a double room (fan cooled, shared bathroom) for the equivalent of 8GBP for two people. Places like Langkawi and Koh Lipe are about GBP 25 a night for aircon and private bathrooom, although the sky is the limit. Food is generally a bargain, wonderful curries for less than a pound.
you do need to do lots of research; visas, safety, etiquette, dress,what to see and how to get around. Also health; some of this area is malarial and another poster has already mentioned dengue which is a scourge in the region. Dengue is not fun and can be very dangerous, so you need to take anti-mosquito measures all the time (the dengue mosquito bites day and night)
Hope this helps; good luck!