Your life should not be dictated by your husband's perceptions of you. Know what? The reality is that divorced people don't send their XW e-mails to read to the kids and expect a response. You pick up the phone and you call them. Your life is not on hold waiting for his next call and e-mail. If it were me, I would stay so busy that I didn't even check my e-mail more than 3-4 days a week and I'd only respond to half of them. That's life. If he has a problem with that...just tell him you've been busy.
As Ian said, you are too worried about what he's thinking or what impact your actions have on him, rather than focusing solely on yourself and your own growth. He's a fully functional adult....if/when he wants to be part of your life, he'll make it clear. If not...still his problem. Take ownership of what you can control.
I am on the computer all the time for school and communicate with professors and classmates via email. He knows this and knows I check my email several times a day. He has been told we are busy--did that around Thanksgiving when he called and spoke to me about D9. He did not seem remotely interested in what busy consisted of for anyone so I did not bother filling in the details. As you said, if he wanted to know, he would make it clear.
SMW
M40/H36 T16/M14 4K B2/08 S4/08 current
Love never gives up, never loses faith, is always hopeful, and endures through every circumstance. I Corinthians 13:7