Here's the solution (long term)
"in order to get, you have to be willing to give"
Everyone is given an opportunity to sign up as potential organ donors (upon death). People who are signed up get first crack at available organs based on the same medical criteria as now.
Same folks as currently administer the program continue.
It won't take long for people to figure out that signing up is cheap life insurance and the supply of organs increases exponentially.
The only problem with that idea is that many patients who need transplanted organs have systemic syndromes and illnesses which degrade/erode their otherwise functioning organs.
Would you want a transplant from a long term drug abuser/alcoholic?