I am 52, poz, have an exwife and children who are in their 20s. You are doing a good thing, keeping involved with your children. It is better to be an involved father instead of a deadbeat dad. There is a difference, so don't let your children guilt you into thinking you are something you know you are not. Two of my child-in-laws (two of my kids are married) have told me how my children told them I was there, with the money, though their mother complained about money. I say this, because there is your story, your exwife's story, each of your children's story, there is the father's story and the mother's story. Does not mean anyone is wrong or someone is right. It is not an easy road, it is worrisome, dreadful, cheery, all types.
Whatever life you are making for yourself, is a good thing. You don't say what life you are making with your children APART from their mother. There are things, to do together, but things, you & she, needs to do with your children, separately. This lets the kids see you are okay, their mother is okay (hopefully), more importantly, the kids feel they are okay. Children will still think they are to blame somehow. Even when you tell them it is not their fault.
I am sure you have guilt, this is normal and somedays, that guilt is feeling unbearable. Here is where you must be careful with yourself. It is easy to give in and do something dumb for quick relief or to ease the pain you are feeling. This is how mistakes are made, whatever it may be.
If the mother of your children is spending her time with the children, while you are not working, then you have time, for yourself. The 'you' time, is where you can decide to pursue this person you have found. Realize, not every bit of 'you' time, will be spent with this person. You still need 'you' (me) time. This is good, even if you are in a marriage, working, with or without children, etc.
I suggest you review the '7 Habits of Highly Effective People' and take from it, the parts that you can fit and use in your life.
As for being on short time, you really do not know this. You could die tomorrow, in five years, 20 years or 30 years. There really is no way of knowing and it is better this way. I have seen people who retired in their early 60s because of health reasons (not hiv) and were given less than five years to live, where they are alive at 83! I've seen some where they died quicker than were told. There simply is no way of knowing.
Do what you can with your life now, watch your food intake, watch your weight, eat healthier as much as possible, take supplements if you need, try not to live in compartments with your life.
For this new person, give it time. There is no need to rush, this is the beauty of a real relationship with another. You have lived the man/woman have kids, life, but life is not this way for many in reality. It only appears to be. This is true no matter the sexual orientation. You are simply one, who has decided not to live this way. Good for you! Don't! But, you will have to rework your view of relationships and what they mean to you and those involved with you.