For Economy Class travel with the points you have, I'd suggest looking at united.com -- no log-in required -- for travel on UA and their various partners. UA does not collect any carrier surcharges on any of its award tickets.
If you can get yourself some AA miles -- perhaps through credit-card sign-up bonuses -- you can also get good deals in Economy Class. AA partners with CX, JL, and China Southern. Again, you can search at aa.com, no log-in required. But for award availability on China Southern, you must call AA.
Good deals on JL Business and First Class awards are sometimes available close in with AA miles.