I have set up the "connector account" thing in Stackdriver, and I'm able to see metrics in GCP, which is super cool! I would now like to also get all cloudwatch logs into google's logging tool. Can this be done with the connector account as well? Or will I have to write a lambda to ship those off?
You can integrate AWS CloudWatch to send metrics to the Google StackDriver. For more information please read these documentation [1][2].
Regarding your Lamba question I found this stackoverflow link that might be helpful. I would also recommend reviewing the following best practice when using a Stackdriver account to monitor any number of GCP projects and AWS accounts.