A very relevant thread for us as we travel annually to Asia (just came back from Taiwan and Thailand) and absolutely hate checking bags; don't want to have to wait at baggage claim and don't want to risk bags being "lost," even if only temporarily. If we're able to book cheaply on a US carrier, we'll do that, and sometimes they will have the best deals. Many times, however, we find ourselves on budget airliners, to include Thai AirAsia, Peach Airlines, and Jet Star, all of which have strict carry-on limits.
As an initial matter, we try to use light-weight luggage; my roller from Walmart only weights 4 lbs and gets the job done. On top of that, we have no problem packing a bit more lightly and relying on local laundromats, which we've used in both Taipei and Bangkok. We've been lucky to have laundromats close to our hotels and have no problem leaving our clothes unattended during the wash and dry cycles. We simply use the wash and dry time to explore in the immediate vicinity.
Apart from the above, we'll often pay for extra weight allowance with those airlines that allow it in order to avoid having to check bags; usually $20 per person extra for what we're carrying, though most of that is due to my spouse's work computers and personal cameras

For those airlines that do not allow people to buy extra weight allowance for carry-on luggage, we'll check some of our things and just deal with baggage on arrival.