IE is partially to blame, it doesn't fully support CSS3 that the new
customization tool utilizes. However, I'm using Chrome, and am also
experience save issues and size issues. I am able to change and save
on occasion though. What you can do for a temporary fix is revert to
the classic customization tool by changing the url from
http://www.tumblr.com/customize/YOURBLOGNAME
to
http://www.tumblr.com/customize-classic/YOURBLOGNAME
Try doing this and see if that fixes some of the issues you're having.
Apart from that, we all just have to shoot an email to Tumblr support
and wait it out