No reason Adams can't be skied enjoyably in early July. I skied it last weekend of June last year and it was still primo - carried less than a mile, skinned to the summit, SW chutes skied great. It likely would have lasted at least 2-3 more weeks in good conditions, longer with more hiking/more suncupped skiing.
I feel like I say this all the time to people starting out in mountaineering, but I believe Adams (and volcano/backcountry skiing in general) deserves a little more respect than it often gets. I don't know what your mountaineering background is - maybe you've got years of experience. If so, feel free to ignore my advice. If not, I think you should be at least a little concerned about routefinding - it's not terribly complicated, but simple mistakes can have big consequences on a big mountain. You should definitely still be knowledgeable and concerned about the potential for wet avalanches. They can still happen in summer - especially on steep terrain like the SW chutes. And you should be aware that the hazards of climbing and skiing a big mountain aren't the same as those of skiing in a resort, no matter how challenging the resort terrain may be. A week before I skied it last year it was an ice skating rink and several people took big, high speed slides with serious injuries. I don't say any of this to be discouraging - it's an amazing ski trip, absolutely one of my favorites, and you should totally have a go at it. Just give the trip the respect it deserves.
Also - no one else has really said it, but I know my feet/legs would absolutely hate me if I tried doing a 7k foot, 10 mile day as my first day ever, in rental boots no less. Maybe I'm a pansy, but a day like that still wears out my hip flexors and threatens blisters even in well-fitting, well broken in boots after I've put in 30+ days and 60k feet in a season. Skinning can either be a really efficient means of travel, or it can be extra exhausting, depending on your technique. I'd take some other folks advice and do a shorter, easier day a little earlier in the season to figure out the mechanics of things, then go out for a fun day on Adams in late June/early July.