Hi, I'd like to help you with your question.
It sounds like you have a couple of things going on here. One, you may still be run down from being sick. You had major illnesses and sometimes it can take several months to feel strong again. Also, you have gone through a traumatic incident with your separation. Not only is your ex not around to be supportive, you are raising two children on your own with no support from him. That is a major stressor in and of itself.
It is no wonder that you are feeling the way you are right now. As a result, you may also have a mild form of depression and a stress related adjustment disorder. Given what you have been through, this is completely normal.
If you can, consider seeing a therapist. You can ask your doctor for a referral or if you attend church, your pastor can help. If you have a local community mental health center, you can go there as well. You may need some short term therapy just to help you with your symptoms and to give you some support until you feel better.
Keep talking with friends and family about how you feel. Talking is the best form of therapy and the more support you have, the better you will feel. Also, you may want to go out for short periods of time to treat yourself to some pampering. A low cost manicure, a new haircut, or even just to see a movie. All of those types of activities can help you gain a new perspective and help you boost your self esteem.
Also be sure to get lots of rest at home. Take time each day to sit and read or just have a cup of tea. I know it is difficult when you are the caregiver of two kids, but if you can try most days to take some time, it will help.
I hope this has helped you,
The children are old enough then to understand your new relationship but both are at an age that their development may be affecting how they see your new relationship.
You may want to consider therapy for them as well. Both children are in adolescence, which already has fluctuating emotions and hormones so adding the stress they have experienced they are dealing with a lot. It would be helpful for them to have an outlet for their feelings.They have lost a father, been through the stress and fear from your illness and now are trying to deal with a new relationship. Those are major life stressors for them and can affect their emotional development. Talk with their pediatrician about a referral or try the same routes I suggested for you. Their school counselor might also be able to help.
My best to you,