It sounds to me as though you're battling some depression, especially regarding the loss of your child to someone you no longer care about.
The motivation for working so you can pay for child support might increase if you were to be able to know about how the child is now, have pictures etc. However, if you can see it from the child's point of view, having a daddy who sends money so they can have things and all, is a nifty thought. I doubt the child thinks that way now, but might later in life when they are on their own. You need to be able to meet with them at that time (of course I hope it's soon) and be able to say, even though...you sent the money to provide for them. Being a good dad means being there for the child even when blocked by another.
Life is more difficult during these days of poor economic conditions. It's hard to have hope. Nothing stays the same forever. Nothing. See if you can begin to make plans for "when" things do change. Set some short term goals, and long term ones too. Find something that gives you stress and depression relief and do it from time to time (even if it takes a little money.) While few people are spending what they used to on vacations, it's still a necessary element for relief.
Yes, looking for work should be a full time job. It's always been that way, for those who need to have income. Try to become organized about what you will do each day (at least 5 days a week) in sending out resumes, going on interviews, filling out and following up on applications, making sure your clothes are ready for interviewing and job hunting etc. It is a job. It pays off later, not daily though.
If you become discouraged about finding work you can enjoy, then figure the stats on your own efforts. Odds are there are some for you to have hope about...for every xx number of applications you'll get xx number of calls for interviews...for every x number of interviews, you'll have x number of recontacts etc. The law of averages, odds and stats will pan out imo.
Try to eat correctly. Many people who are down in the dumps eat like they are digging through the trash in the dumps. Without proper fueling of the body, it won't work right, and you won't feel good and it will also effect your thinking. If there isn't enough money to eat good each meal, then spend some of it on a bottle of complete vitamins.
When reading your post I was reminded of how I lost 5 jobs in a row, each one lasting less than 6 months. It was not my fault, indeed, that's the truth, but still I was the recipient of the bad record. I had to let go of the grudges against the previous bosses and go out looking with full intent of still giving the job my all, and holding true to my principles.
Do you really think it was depression or laziness (or a combo) that caused you to lose your jobs? Make a list for each job you held, what was good and what was not good, and where you may have gone wrong.
Is there any chance of returning to one of them and asking to be rehired? Being able to talk to them about what you would do differently might help (if you can.)
Good wishes.