I would approach this as two different problems.
1. Check the Minimum Connecting Time for LAX intl to domestic. I think 52 minutes is doable as you will arrive and depart from the same terminal area (6&7). If you were coming in at the Tom Bradley Terminal I'd say forget it, but 52 minutes isn't disastrous (unless something goes wrong of course). For your peace of mind (or if you see Saver space in F on another flight) you can try to change it, but you may have to hang up call back again to get somebody who wants to change it, if the MCT is 45 minutes. If it's 60 or greater, you have every right to change, and there does NOT have to be saver award space available necessarily to confirm a new flight.

2. If you booked a C/F award, you should clear the waitlist, even if it's at the gate. Status would probably help, but you can, I believe, worry about this later. Because you're seated in a lower cabin than you're ticketed in, you are near the top priority to be put in F, regardless of status.