In my opinion Lead Alto is one of the most beautiful requirements of the horn:
You have to have enough warmth and full overtones to blend with a section,
you need to be able to go pp to ff without much effort,
with some "gas" you need to ride slightly on top of the Sax section - and -
with even more gas you need to be able to blow over the entire band during a solo - and the band could be screaming. (Those damn trumpets!) lol.
The best mp design for this IMHO is a roll-over baffle. I hate the sound of high-baffles in most big band contexts - too much edge, too many high overtones, few low overtones or "body." For instance, I would never play my metal Dukoff D7 or 80's Beechler in a big band context on Alto. MIGHT work on a lower pitched horn like tenor or bari, but not Alto.
For the classic Big Band Sound, I would go with a rubber Brilhart, Meyer, Link, Lamberson, etc. as well as some of the "throwback" designs by CE Winds, Jody Jazz, etc. Look for a mp with an unrestricted throat (get some AIR in the horn), curved or straight sidewalls, NO raised baffle and a slight convexity rising towards the tip, starting about 1/2" away. This is the "roll-over." This slight raise, when done right, is pure MAGIC for a great Sax sound, regardless of the voice (SATB).
The poster who talked about Diaphragmatic Breathing is right. If you're not doing it now, LEARN how to do it, all the time. Power comes from your belly, not your throat or chest. It's a "huff," not a "hee," if that makes sense.
As for pitch, another poster hit it right. The old saying regarding Lead Alto used to be "You can play sharp or out of tune." When I first heard this as a kid, I was confused. Then I did Lead Alto work and understood. You can be spot on pitch and sound fine... but you can also be SLIGHTLY sharp and give the whole ensemble an edge. But if you're flat - or if some idiot next to you is sharp and won't tune down, the whole section will sound deflated. lol. You have to get with your section and learn to work together for an agreed aesthetic.
And then there's Timing, which separates the men from the boys. There are zero modern bands that drag like the Basie sax section, especially the 50's Atomic Band Period. But if you can get the section to try it, it's amazing. But it doesn't work on ALL material. For instance, I wouldn't play That Jones' "US" this way. In a driving chart like this I'd actually lean forward in time a little - again, to give the ensemble an edge.
There's a lot to playing in a Big Band. Most of all, listen your ass off and HAVE FUN with it!
Oh, and definitely find a trustworthy mp refacer to check your mp. Whoever said it's a waste of money is either very lucky or ignorant. I can pull a dozen name-brand mps off the shelf - new - and show you many of them have warped tables and crooked facings. Just because it is stamped "Meyer" doesn't mean anything. Every piece is different. The day you take your "it almost plays well" mp to be worked on and feel the difference, you'll know what I mean.