Hi Franz, welcome to the site.
Congratulations on 3 months sober, though it sounds as if you are drinking now. If that is the case I hope you can take a moment to stop and re-think that decision.
Your Visa situation sounds like a mess. I do understand very well how risky it can be. I know of a young man from India who remained inside or at least on the property where he was living for over 5 years because he did not have his papers in order and if caught would have been in a lot of trouble. I also help out a group of refugees who are here without any official documents and while we have it sorted now to where they have some sort of ID and can thus be out and about, they are living very much "on the margins" of society and it is a really disheartening situation for them. We have endless appointments with the government officials and lawyers and it can get so frustrating at times.
I am not sure what country you are in and which one you are from, so it is hard to offer specific advice. However, I do know from my own experience getting a visa and helping these other people that you just kind of have to jump in and start. There is so much paperwork and photographs and fingerprints and documents and this and that but if you slowly, each day, make some forward progress eventually you can get to the end of it. For example the Indian man I mentioned above, it took him several years but he is now completely legal and official, has a car, has a daughter in school here, leaves the house whenever he wants. I have to re-new my visa every two years and it is a huge headache, but totally possible.
Another thing to remember is that "no" is hardly ever no. There are a series of appeals and processes you can go through to fight. Also, having children there is a big factor in your favour. While you don't want to be out and about with absolutely no paperwork or permission to be there, once you start the process you start to have more freedoms. In addition, just having an open case of appeal buys you some time, Several of the refugees I help are living that way. They get a "no", appeal, and then have another several years of safety. It is not idea, but can buy you some time. It is very highly unlikely the government of where you are is going to collect you, pay for your airfare to send you back to where you came from. Think about it that way, it would be an incredible amount of work and money on their part to send you back. In the end it is easier for them to shuffle through the paperwork of your application.
All this said, it is a LOT of work and a lot of stuff to organise and remember and dealing with it while drinking would make it enormously more difficult.
I really feel for your situation. it is scary, sad, frustrating and you feel like no where is home. But this is no way to live. You can find independence and freedom but only if you start putting one foot in front of the other and move towards a legal, legitimate life where you are.