Yes it could be the flex disk, but the diff mounts are common to fail because many people (been there done that) do not know that the diff is rubber mounted so they pick the rear of the car up by the diff, ruining the mounts.
I have never replaced a flex disk anywhere near as soon as 60K miles. I suppose if you really drive hard, it might last no longer than that.
Diagnose before you replace anything. If you were to put the back wheels on ramps and then lightly raise the diff with a floor jack, you will easily see whether the diff mounts are bad or not. To check the flex disk, simply grab the driveshaft near the disk and shake. It should be tight. Don't forget to do this at both ends of the shaft.
Replacing the diff mounts takes much more than bolts and washers. There is a special puller that is used. My brother in law made one that worked great, but he can make anything on his lathes or mill. If you don't have a lathe, you can buy a puller from Performance Products for $110. I think they also rent them.