Hello Do Do,
your Visa from the Netherlands don't matter any more after 3 years, if we speak about responsibility of another EU-country for your case.
To speak about a possible asylum application or any application for humanitarian protection it is important to know what country are you from and what problems you have there. That means: why you think you can't go back, if you really have reasons, other than your wish to stay with your boy-friend. An asylum application can naturally be a method to legalise yourself in Germany at least for some time, to get a document for the time of your asylum procedure and to use this time - for marriage, finding job or professional training (Ausbildung) etc. etc. You must take into account that there is a distribution system in a German asylum procedure and it is possible that you will be sent to another federal country (Bundesland), far from your boy friend. It depends of your country of origin and on the quotes for the federal countries. Another problem is that you sometimes must stay in an asylum camp for a long time, with various limitations. Nevertheless, it can be a chance, if you need time, for example for preparing your marriage. Absolutely individual.
If you really want to get married, you must first see if it is possible to collect ALL necessary documents. The rules for a marriage are different in different countries, so that it is often much easier to get married in your country of origin and then to legalise this marriage for a family reunion procedure to Germany. Or to get married in another country, where you will stay legally at this moment. Many people use Denmark as an option but you must come there and stay there legally to do this.
I know that if you have ALL documents for a marriage in Germany and you already have a near appointment for the ceremony in Standesamt (in a couple of weeks), you can apply for a Duldung because of that. But I only would do that after consulting a good lawyer with expertise in migration law (necessary!), the best is to take a lawyer for supporting you, because coming out of illegality is not an easy thing, you can't be sure that the Ausländerbehörde won't start deportation measures. You can also count with a criminal procedure and a money penalty.
To get a residence permit in Germany on the basis of family reunion to a German person you must have A1 Certificate German language. To cover all costs of living is not necessary if the spouse is German but it would be an advantage in this situation. And even after the marriage the Ausländerbehörde can claim that you leave Germany and make a proper family reunion procedure from your country ( taking a visa from a German consulate).
If you want to get a legal status in Germany for work, professional training of studying, you normally must ask for a corresponding visa in your country of origin or another country where you stay legally. If you have such a possibility (a good job offer etc.), you can think about this. It would be very important in this case to leave Germany volunteerly, without getting a ban on return. It would be difficult to get a legal status here only because of these aims without further humanitarian reasons (like for example a disease which prevents leaving Germany, pregnancy etc.).
It is not possible to evaluate your situation in any sufficient way without knowing a lot of details about you (age, education, language skills, job perspectives etc). Online counselling is not enough anyway. As Meike recommended, look for a good councelling center for migrants in your area or a lawyer with a corresponding expertise.