My 3 (2m 1f) are bilingual English French and do well in German. The two boys are out of school into apprenticeships and the girl is in secondary school, so had lots of years of this...
When they were small, they would accelerate in one or the other languages at will - no rhyme or reason.
After they started school when they were playing they would talk in French then, mid sentence, change to English to talk to us and then back to French between them like nothing had happened.
We were teaching our daughter to count and she was happily counting from 1 to 10 in either French or English. When someone asked her to count in French - she was confused as she had thought that the numbers had two names each, no idea that it was to do with two languages ...
Reading took a hit for a while compared to her classmates, but she caught up... basically they are sorting twice the vocab compared to a kid learning one language. Once it gets sorted, and don’t get worried - you need patience, it all turns out fine.
Last year she said she was worried as she had never read out loud from an English book, (fine in French). I said don’t worry let’s do it and gave her a Harry Potter book. So lots of errors but she was amazed she could read English and has improved since then.
Go with the plan in your first paragraph, but if you, or your partner, strays into the other language don’t worry.
EDIT: if you join a conversation, then join it in the language it was started in, however, if you swap to the other language because it flows better or for any other reason, then do it - the kids will understand anyway and this does help with associating words across the languages.
We found that whenever they ask for a word or explanation of a word in one language to give the word in the other as well and similar words as well. Just go for it.
Watching tv with subtitles (VO etc) is also good fun - discussing what and how much was skipped by the text - amazing how much you can read and hear and process the difference at the same time.
Just wait for when they pick you up on pronunciation or vocab in your second language :) been there had that.. :) It’s a fantastic sign that you did it right.
Oh, if you are living in Germany, just wait when the first one comes home from school having corrected the English teacher... Last time this happened the teacher decided not to meet me, but took that question out of the exam - number 1 son had pointed out her error in detail...