There isn't much you can do. The transition guide blithely recommends copying and pasting but unfortunately pasting formatted text into a text editor is rarely a good idea. You should really paste to notepad first to clean out the OLSB formatting and then format using the 365 tools. It's an extra step or 2 but may save you some of the aggravation you're experiencing with the 365 editor.

This should have been included in the instructions but apparently MSFT didn't test a real life transition out. 

 

Hi Jlo3603,

This is Jonis from Microsoft Online SharePoint Support.

While I was not directly involved in the OLSB migration, I can recommend the information provided by mch.  It has helped a number of OLSB users who were migrating their information.

I would also recommend that every time you finish editing your public facing web site you:

1. Save your changes and publish them to the Web.
2. View how your website will look to visitors.

By doing both of these on a consistent basis, you could avoid design change irregularities.

Please reply to this post with your results as I will continue to monitor this thread for a couple of days and will reply to any additional posts or questions.

I am also rebuilding from OL. And although I did paste the text into notepad first, it still won't format correctly. I am pasting text into a table cell and it won't un-bold. I tried recreating the table, pasting into different cells - nothing works. Alas, no aggravation spared with the dismal 365 editor...

If you're pasting into an area that you originally pasted directly into, that can happen because residues of formatting get left behind even if you delete (which is why pasting directly isn't a good idea). Try sweeping the entire page with the right  LEFT mouse button clicked and deleting everything. Like OLSB it's  a very difficult editor to work with.