I’m not 100% sure on this in the case of AGPL, but I think you don’t need to relicense your project if you include AGPL code; you only need to make sure your project respects all the freedoms the AGPL requires it to (in a suitable way).
So your own code would still be under Apache, and people could follow only the Apache conditions if they only use your code. But combined with the APGL part, the project as a whole would of course have to follow the APGL conditions.
GPL and AGPL typically imply that your entire project is licensed under those conditions is my understanding. I find it silly to licensed something MIT or BSD but pull in some GPL code, since now the entire thing needs to comply. GPL is about end-user freedom by force against the developer. Don't get me wrong I love the GPL, but if I want to use a specific license I rather stick to that license.
> you don’t need to relicense your project if you include AGPL code; you only need to make sure your project respects all the freedoms the AGPL requires it to (in a suitable way).
correct