This is going to be an essay because it is deeply personal to me having been through something very similar several years ago and my advice and reaction goes completely against most MN thinking. I had to go through the complete grief cycle to get to a point of acceptance before I could get ready to rise again. That took me many months.
@Zebracatis spot on. Regardless of whether it is for being alone or with someone else. You have to do a full 180 from the despair and despondence. Think of it as pushes and pulls. At the moment, crying and weeping and begging is pushing him away, Who wants to be with someone who is behaving like that? Simply put, it is not attractive. You are (whether he would admit it or not) in the way of his 'happiness', you are blocking his fantasy of skipping off carefree into the sunset. Get out and start living your best life, even if it feels fake to start with. Make yourself unavailable to him, too busy. Keeping busy has the added bonus that you don't have so much time to feel sorry for yourself.
There are no guarantees but, changing your behaviour is far more likely to make him curious, and want to understand what changed. He is coming and going from the house, checking on you (again, without having to recognise he is doing it), and if you are always there wallowing, he doesn't feel the shift and thinks he holds all the cards. By finding yourself and your power, you become a lot more interesting and far more likely to pull him back. IF YOU STILL WANT HIM BY THAT POINT. You may well decide you don't, but you will be in a much stronger position to look after yourself either way.
Now for a brief(ish) summary of my story.
I was that woman crying, weeping and begging, losing weight etc. It took me a LOT longer than 5 weeks to pick myself up, turn myself round and get going again. After that, I made new friends, tried out new activities until I found things that stuck for me. I bought clothes that fitted my new, substantially slimmer body, started going on holidays by myself, learning a new language, bought new furniture, rearranged things my way etc. The way my counsellor put it to me was to work on four things - body, mind, soul and spirit, and to do something for each, every single day. Some days, it felt completely insurmountable but, I ended up on a voyage of self discovery like I had never done before. One that I continue with even now.
My husband would be turning up at the house and be all surprised that I wasn't sitting waiting for him. After one such incident, I watched the realisation dawn that he no longer had any insight into my life and that I wasn't the person he believed me to be. He had thought I would always be sitting waiting, and he did not like it when I started moving away from him, not one little bit. He broke off his other relationship and begged my forgiveness. We are now back together after a year of complete separation and barely speaking followed by a year of dating for me to decide what I wanted. Ultimately, for me, it was still him, and I never felt I was playing 'pick me'. I have no regrets about my decision, precisely because it was mine to make by that point. It is a cliché but now he is the one working not to lose me again.
One note of caution, I got a lot of legal and financial advice along the way. Early on, I had decided that if he really wanted out of the marriage, he would have to take ownership of that process. I got told multiple times on MN that I was an idiot. Yet, all the solicitors I spoke with agreed that, in my individual circumstances, there was no financial imperative, so there was no pressure to do it myself. That will not apply to everyone, and protecting yourself has to come before anything else.
I have name changed for this and will do so again afterwards as every time I post fragments of my story, I get vicious personally attacking messages. If anyone reading this is tempted to do that, please don't waste your time.