I rent with other adults in a rental property and my name is on the rental agreement with them. I would like to know if I have to move out, or is there a way I can remain and still pay just my fair share of the rent, without my credit rating getting harmed if they do not pay their fair shares as well.
I've lived in a few shared houses in my time. One house had one unemployed guy, two students, and two of us earning very little. We had individual rental agreements with the landlord, so if one of us missed a few payments they'd be liable for eviction without affecting the others.
To be honest I've not heard of this arrangement elsewhere. Every other rental agreement I had we signed a joint contract. I've had to pay housemates bills a couple of times to avoid a judgement against us and it was a pain getting my money back and strained our friendship. I'm still owed a few hundred that I'll never see again.
In general if you can't trust your housemates to pay their bills you're probably better off looking for somewhere else to live.
Per your comment that your landlord is a large property management firm, I think your credit is definitely in jeopardy.
Assuming that your name is on the lease, there's no distinction of "fair share", you're all responsible. So, if your roommates can't pay their share, you might have to pay it for them, break the lease (more money, depending on terms) and get out of there.
Alternatively, you might be able to find new roommates who would be able to pay their share(s).
Unless you can make an agreement with your landlord, your credit is on the bubble in this situation and it may be difficult to get the landlord to void the terms of the original agreement in lieu of a new one. I was burned by a similar situation when I was in college. I rented an apartment with my then "best friend". I sent my half of the rent in on time and he consistently skipped out on his. My credit took a blow to the tune of the shared liability just the same.
Seriously seek to work something out with the landlord because it sounds like if things continue as they are, whether you move out or not, your credit will take a hit.
In San Francisco (home of super-expensive real estate and more than a bit of marijuana use) you'll see people asking up front for "first and last months' rent" plus a security deposit when you move in somewhere. If someone can't come up with the cash, then it looks like this is the last month... and time to look for a new roommate.
of course, it can be difficult to implement this arrangement if you've already moved in with a bunch of people, and if it's not common in an area people may balk at the money required up front, but.... it is a reliable way to deal with people flaking out on the rent money.