I can only speak for the OPC, though I don't know of any other Protestant denominations that issue annulments or certificates of divorce, either. That said, if you divorce someone for unbiblical reasons, you WILL come under church censure that could lead to excommunication if you remain unrepentant.
Also, if you are married to a professing believer and wish to get a divorce, it is expected that you will file charges against them with the church where you are a member. Only after the discipline process is complete, and your spouse is excommunicated after being found unrepentant in a sin that biblically allows for divorce will you be permitted to file for divorce without any negative repercussions, such as church censure against you.
As for annulments, I would need to hear examples of what you mean by unbiblical marriages. Like, marriages to a close relative or someone of the same sex? That would be examples of the marriage itself being sinful, and therefore part of repentance would include dissolving the marriage, which is no real marriage at all.
Personally, I also would prefer to see the State entirely out of the issue of marriage and divorce, but I also recognize that the State is often needed as a mediator in the case of distribution of common property, which is not properly within the purview of the church.