This does sound like a very tough and challenging situation. I recently read a book by a dad in the same boat, he had a blog called https://lifeasawidower.com/
You are not alone! Nobody can do it on their own, 24/7, all year round. It is good you have the support of your sister, but I believe your kids would probably miss you and their home if they were to live with a new carer and not their parent.
Please keep reaching out to all the help that is out there. Have you joined WAY (widowed and young)? My friend lost her husband too when their kids were young, do keep building your network, take your meds or see the counsellor, whatever works for you, be patient. This is a horrible, crisis situation and you are surviving as best you can.
You are the only dad/parent your kids have, hang in there and I am sure your network, your strength, your confidence will increase over time.
There may be a HomeStart service in your area, a volunteer who comes once a week and offers support. Speak to Camhs and your kids’ school/nursery, there is definitely help and you don’t have to master this situation on your own.
We all lose faith sometimes, and struggle with low moods. You never know, your ex may come back into your life and understand you better, and you may not be alone forever. Go to some Gingerbread group meetings and have faith in yourself, I am so sorry for your loss.x