If the DP cable doesn't fix the problem I'd try to use or borrow another graphics card that DOES have DP, or swap in a different monitor that has DP, to eliminate one or the other, because it COULD be a problem with the DP circuit on your display as well.
If you try another graphics card and it does the same thing, then you know it's the monitor. If you try another monitor and it does the same thing, then you know it's the graphics card. And in both cases, visa versa.
In many cases when it comes to troubleshooting hardware problems there simply ISN'T any easy way, you HAVE to swap parts in to identify the issue. That's just the way it is.