Need advice on how to change job

I posted about this situation a while back on reddit and got really good advice. Here's the situation: The company that I work for has put me on Leave without pay since November last year. And so I posted here asking for advice and as per them I started preparing for job interviews. Now I'm getting interviews and calls from recruiters, but when asked about my resignation, I say that I've resigned and can join immediately. Now since I'm on LWP I can resign immediately and join the other company, but I haven't resigned yet. So if I clear all the rounds and they ask for salary slips and resignation acceptance letter and other documents that date would be a more recent one. So for the resignation part what should I tell the recruiter, something like I haven't resigned yet and my notice period is of 30 days but I can negotiate it to two weeks or something else. Another option is that I can tell my current company to put an older date of resignation when creating the letters and documents. Also for the salary slips of November and December and now currently January it would be blank because I'm not being paid. What do I do in this situation when asked for salary slips? Should I for example say that I resigned on November and show salary slips till October and tell my employer to put a data of November 2024 on the documents, or do I explain it to them in a corporate lingo. I need to form an answer that would be plausible to handle both these situations. I really need help on this matter as I honestly have no idea how to handle this. Thank you so much for reading my post and for helping out.