No reason you can't do both. Laos takes about 5 days, Burma 7-8 (will the "it takes 3 months to see a country" crowd please shush!)
Forget the rainy season problem, you are going when it's too **** hot!
For itinery ideas, take a look at couple of the many package tours which cover these countries in this timeframe. If you don't want to it it on your own, private guides add very little to the cost.
BTW: Laos visa available at the airport/border. Gotta get you Myanmar visa ahead of time.