Hi Jillian. I'm not sure if the google docs are embedded through the LTI of if they are linked, so I'll comment on both:
LTI: For any LTI shared assignment (even if you reuse it yourself) the Google doc must be reconnected after the share. (A Canvas support person gave me the technical details, but I couldn't really explain it. )
To share with a colleague, there are lots of options, but one easy way to share course materials is to create a shared Google folder. Each of you can save files to this folder for embedding in your course.
LINK: All Google resources for the course must be shared with the other instructor. This will allow you to share the course without any edits needed after the import. You can use the share button in any doc to grant rights to the other teacher, but again, a shared folder might be a good option because you can set the sharing rights on the folder and new materials will inherit those rights. (Moving your items into a shared folder will not effect the links in your courses. No updates needed.)
HINT: Be sure both of you have a solid understanding of how Google file sharing and link sharing properties work. Mistakes and misunderstandings can result in other teacher and even students editing your course materials.
I noticed that we have not heard from you since you first posted your question here in the Canvas Community on June 2nd and then received a couple replies from both firstname.lastname@example.org and email@example.com later that day. Have you had an opportunity to review these responses? If so, did either of them help to answer your question? Or, are you still seeking some help with your initial question? If you feel that one of the above responses is "Correct", please go ahead and mark it as such. However, if you are still looking for some help, please come back to this thread and post a message below so that members of the Community can continue to assist you. For now, since we've not heard from you in quite a while and since there hasn't been any new activity in this thread, I'm going to mark your question as "Assumed Answered". However, that won't prevent you or others from posting additional questions/comments below. I hope that's okay with you, Jillian. Looking forward to hearing from you soon!