Why do you need to switch your style?
Most things get better by practice so if you really want to do it, then find some things that would at least theoretically work well together (such as watching the show and writing a blog post or reading a book) and start doing it. Keep at it for minimum of 1 month. You should find that you're slowly starting to get better and maybe you might be able to start mixing and matching more things.
Alternatively make a schedule that works with however your brain wants to work and try to cut down on things that aren't important to you (no matter how much other people scream that it's a matter of life or death to them that you do it) to make time for the things that are.

Actually, I was thinking anti-multitasking thoughts throughout the day.
I really do think we are giving multitasking way more credit than it really deserves. Ok, It may fit the requirement here and there, but not always.
I conclude that it is a 'nice to have' skill and not a 'must-have skill'.
This really should be my primary focus. I think I spend too much time doing things that need not be done at all. Like how I write and keep editing it to a point where it's a completely different story altogether. The narrative gets changed and it's like rewriting the same thing 100 times.
Ahhhh.!!! Too much pain...
I have a combination ready to test... (Audiobook + Workout) It worked in past but lets see if it still holds strong. XD
I must tell you that feedback from you is the best part of my Hive experience. And you have inspired me to share this experience with others. I am checking out random posts and commenting on them. I am keeping the number to around 10 per day for now. But I am going to take them a notch up :D
GoofyFriends
If it's any consolation I think it's a lot to do with how society has been going. Like for a while there seemed to be a trend where bemoaning how exhausted you were was a badge of honour as it was "proof" that you were "productive". And then because people are cramming way too much in (because YOLO and "need to be productive every waking second" "doing anything possible to prove not lazy" whatever) "multitasking" becomes the way to cram everything and more into what time you have.
If it's a thing that comes natural then it's an amazing way of actually getting more done. If it's not or is something that you're currently working on, you may just end up feeling constantly burnt out and resentful and not actually be getting more done at best or even less at worst XD
Amusingly your ability to concentrate on one thing at a time and absorb it in its fullest is probably the goal of a lot of the people practising mindfulness.
Good luck with the audiobook + workout :D