Best I ever had was in a 1968 prog (big surprise...) I had been building this team for 1968 for about 5 seasons, specifically because of the quality of the SPs on the team. Rotation of Gibson, Tiant, McDowell and tandem of Barber/Ryan in the 4th spot. 33 shutouts on the season, allowed 2 or fewer runs in 88 out of 162 games. The team allowed 456 runs total (about 2.8 per game) and had an opposition BA/OBP/SLG of .201/.284/.278.
I started a scrub the last couple of games of the season to rest those guys for the playoffs, otherwise the opposition BA would probably have been under .200.
The daily scores for that team are no longer available, so I can't determine what our longest SHO streak was. I don't remember having 5 in 6 games though, and I keep pretty good records for my prog teams so I probably would have noticed it if it had happened.
Sounds like you've got a good looking club there; GL the rest of the way.