You can use multi-targeting to move parts of your code over to .NET Standard. So if you start with your base projects and move your way up, gradually your code will become compatible between .NET core and .NET Framework. This will help you remove any .NET Full Framework / Windows specific bits. Where you have to have things that are specific to each OS you can then just configure your DI so it will inject the Windows specific / Linux specific implementation.
For those that are not aware of multi-targeting here is a great article on it:
https://weblog.west-wind.com/posts/2017/Jun/22/MultiTargetin...