A fixed week owner still has to pick a check in day so yes they can
Actually, I think that is too late. Here's how I do it and I've never had issues. I start dialing the number around 8:55 am EST and keep hitting re-dial until the line picks up. I also log-on to my computer and hit refresh until the week is available. Lately, I've noticed the online part is a heck a lot faster but you can only do that for 12 month reservations. However, since online reservations have been available I've discovered at the resorts I own at they never book up. Waiohai has availability for every week next year up to this week. Now, not all check-in dates are available but there is at least one day available. So, I think, if you really want that July week and then, at 12 months you are going to have to have multiple phone lines going and the computer to try and snag a week.
I do the same with camping spots. It is the same game every where. I even discovered how unfair the game is at reserving Yosemite weeks. At least with Marriott, there are rules.
So, my question now is that can a fixed week owner reserve multiple weeks? From your post it sounds like all of July has been reserved. So, do fixed week owners get to start reserving a week earlier. Does that make sense? Because if that is the case, I don't think that is fair. I don't think Fixed week owners should be allowed to reserve multiple weeks.