The first thing I would do is begin to make a home for your children. Rather than an apartment, search for a smaller, reasonably-priced home to rent you can fix up real rooms for your children, take them out with you to buy posters, etc to decorate their spaces with, apartments always feel temporary and misplaced. Then, if you do not already have it, obtain joint custody and instead of just a Friday or Saturday night with them, begin to have mid-week dinners in addition to weekends, even if it is just picking them up for dinners out, hanging out at your new home playing games, start to focus on the kids and make your time with them something to look forward to as fun. Be the one to take them and pick them up from karate, gynastics, what have you, be there for teacher's meetings, doctor visits, all of these things that most men never take part in, BE there for them in this and you will find an entire new aspect of relationship developing as you become the go-to guy in all areas of their life.
You are lonely right now, but the real, safest way to deal with this rather than trying to slap some woman into a big empty hole is to become the kind of father your children want to spend time with, feel very safe and cared about around, and most importantly -- that they are NUMBER ONE with. Always. It is heartbreaking when a family is broken up, but I can tell you as a single parent with my children much older now how very solid and loved their growing up years were with me, a parent who put them first with no regrets and have seen them grow into secure, bright kids with a true appreciation for the love and laughter our home always held for them. The same kids who always want to spend free time with me as it always revolves around the joy and love and laughter I invested so much time into being sure they had when they were with me.
You do not need a new family, you already have one. The biggest mistake many men make is they decide they need to re-create their lives and sacrifice their children in the process. DON'T DO THAT. And if you really feel as if it is in the best interest of the kids, get 'custodial' parental custody. I know many people with joint custody who split the households one week here, one week there, I do not like that idea as I think it is disjointed for the kids but there are many options besides just being a part time Dad. Look into them.
Get off the pity potty, you are hurt, it will take many years to heal, but investing that emotion into the kids as the focus rather than self will greatly assuage you in healing and filling that big hole you have right now.
Get off the booze, you will never be someone your kids want to be around if you drink yourself into oblivion every night. Don't be the sad sack, I see so many parents who get divorced put their kids in the position of pitying them. It is the children who right now need YOU to step up to the plate, put your own sadness to the back and put them and their happiness to the forefront of everything. There are many ways to heal, find some constructive books, classes, people to be around to help you through your sad times, but make sure you do not drag your kids into the hole with you.
If you focus on all the possible goodness you can do for your kids at this time, it will go far to fill up the time you are away from them.