Bay County didn't have such as thing as a "marriage license" until 1870. Until then, anyone wanting to marry went to the JP or a priest/minister and just got married. If the County has a record on file, it would be from the JP marriages. The fact that they don't have one says your subjects must have been married in a church.
In the case of the record you found, there's a notation in the corner that it came from a family Bible, along with the email address of the person who has the Bible. I'd start with an email there: firstname.lastname@example.org
From there, since you have the date of 1868 the number of places where your ancestors would have gotten married is very small. So start by contacting the handful of churches from the era and seeing if they have a record of it. If you know what religion they are, start there. But keep in mind that not every religion was represented in the small town in that era, so they may have worshipped in another faith's church. Methodist, Presbyterian, Lutheran and Catholic were well represented. There was one Episcopalian church, but it wasn't easily accessible to everyone.
You asked if they might have married in Saginaw. It's possible, but not probable. From the records we have, it appears that was a very long journey. In the 1850s, it was a 3 day journey. In the 1860s, it was a full day's travel. It didn't become an easy journey until the 1870s.
The other thing you have to understand about "Bay City" is that it was actually several separate towns in that era. When you find records on them, see what it lists as the town where they live.If the records say "Bay City" or list a ward, then they were on the east side in the area from downtown east to Carroll park and south to what is now Kosziusko (22nd street). If it says "West Bay City" or "Wenonah", then they were on the west side of the river. "Banks" is also across the river from what was "Bay City" proper, but it was along a bend in the river, so a little more north than west, with it being the piece of land at the mouth of the river, leading into the bay.
The farther back you go, the different names you'll find for the same land. Originally, all of the area was called "Hampton". Now that's a township east of the city extending to the Tuscola county line along the bay. That was broken up in the late 1850s and early 1860s to form "Portsmouth" (there being both a town and a township by that name for several years. The town was later annexed and is now called "the South End".) Monitor, Bangor and Salzburg were all towns farther inland from the river and bay. Bangor the town is now part of Bay City. The township continues to be separate from the city. Monitor the town has disappeared. But the township continues westward from Bay City towards the town of Auburn. Salzburg has also been absorbed into Bay City, but part of it continues as Frankenlust township.
So as you're going through records, you need to have a timeline in your head of the changes so that you can understand where they lived and what part of the town they were living in. That also gives you a clue to their ethnicity and the culture they were in at the time. Bay City remains very segregated. But 150 years ago, it was EXTREMELY segregated and the different cultures didn't mix very well.