I think I wouldn't include the 1996 songs into that category. According to the info I have, I Get A Rush, Someday Just Might be Tonight and River Runs Dry are from 1996. They were probably recorded with the band but I doubt thta it was done with a purpose of including them on the next album.
I read somewhere that River Runs Dry was the next Always Jon wrote but it was after the album (These Days) had been released and it didn't fit Destination Anywhere.
I have a list, where even Out Of Bounds was listed as from 1996 but I think it was from 1992
And quite speculative are the recordings from 1997 - Shut Up And Kiss Me and All I Wanna Do Is You, the first is confirmed to have been recorded with the Big Dogs but the latter is not however it's expected given the year.

I find it interesting that the songs which were used as b-sides weren't re-recorded by the band. Apart from Jon a Huey, those are totally different people playing. And Real Life was, with the intention to release it as a single. I'd love the band's take of Stay for example.

I love this era and I love Crush the way it is (well apart from She's A Mystery..). I'd love to have Jon's Sex Sells album and then the band's version, which would be the same to some extent with a few Richie's songs added.