You will get less range out of heavier BB's. (However they are more accurate)
Stick to the .20's they are your best bet
Sorry, but I don't believe that's correct.
If you use a heavier bb, and have a hop up, the maximum potential range using a flat trajectory will be increased. This is because you are able to apply more hop-up effect whilst still retaining flat flight. Whether overal range would actualy be increased if you compared the two weights with the same amount of hop, I'm not sure, (and atm. can't be bothered to do the equations to try and work it out) but you will get a greater USEABLE range (flat flight) with a heavier bb.
Also, as you say, the accuracy will be increased, largly due to the reduced effect of wind.
Having said that, the velocity will be reduced with a heavier bb. But as E(kin)=0.5xmxv^2 , a quick re-arange shows that m is proportional to the inverse of v^2, so an inrease in m will result in a smaller decrease in v. A quick example, if you have a gun firing a 0.2g bb at 300ft/s, and you double the bb weight (to 0.4), the velocity only drops to 212ft/s. So where the weight doubled, the velocity only drops by less than a third.
Hopefully, some of that made some kind of sense, but it is late and I'm tired.
The crux of it is, a heavier bb will be slower, but provide better range and accuracy. So it's all a compromise.
If you don't believe me, try it =)
Edit; however, if your shotgun is running at 350 or under, I would be very weary going heavier than .25 as the hop may struggle and the bb's flight will be painfully slow to watch (and thus easier to avoid)
Seccond edit; coherency