I think it is amazing that you are already seeing benefits of the program! It can be tricky at first to challenge negative thoughts but with practice it starts to become automatic. I encourage to try to thought challenge and complete the applicable exercises daily or as often as possible. In time, you will notice a difference.
You ask some great questions. For one, you mention running from feelings. That is not the intention of the homework, but I can see how you might see it that way. With the program, you are trying to challenge negative thoughts, so that you can see things positively and accurately. Sometimes things are clearly upsetting whether you use CBT or not. For example, if someone's spouse dies, you could use CBT, so that your perspective is as healthy as possible. However, it is still sad and feeling grief is important and unavoidable. Some things are sad no matter how you look at it and that is OK. If you feel you are running from your feelings while doing CBT exercises I encourage you to mindfully allow the feelings to come. What sensations are happening in your body?Where do you feel the pain? Etc. Make note of how you feel when you mindfully experience the emotions vs. when you try to challenge them. I would love to read what you experience is with doing both exercises, mindfulness and CBT. I suspect you may find sometimes mindfully allowing the emotion works best, while other times CBT works best..
I am sorry to read you are struggling with self care. Unfortunately, this is common with depression. Remember that it is temporary. Try to set small goals for yourself each day. Make note of how you feel after you do practice self care. I also encourage you to talk to your doctor. Your doctor can rule out other health issues that may be causing you to feel this way and they will be able to explain your options in terms of medication. Studies show that Cognitive Behaviour Therapy works best when coupled with medication.
I hope to read more from you soon!