You shouldn't have this problem. Most people before they start AJATT already have a passion for otaku content like anime, manga, ranobe, or other types of Japanese media. For them learning Japanese is a means to get more enjoyment out of the media they are fond of. If you decided to learn Japanese out of the blue, and you don't like any of the above, it's going to be very hard to immerse without quitting and going back to English.
If you like anime, there will be no temptation to watch crap in English. Anime exists only in Japanese. Most of the stuff people like in Japanese has no alternatives in the English world. So you can either consume translated media which is always worse, or you can learn Japanese and experience the original to its fullest. A lot of stuff isn't even translated, and isn't going to be.
Find something you like in Japanese. Try to discover new stuff, something that has no alternatives in English. I feel like most people reading this already have an answer.
If you look at this from the opposite side, there's no difference. Let's say you're Japanese, and you're learning English. There is unique, interesting English content. You just have to find it, find what you like.
When it comes to should you go cold turkey, you can't do it until you find media you're passionate about. When you do, I bet you're going to jump cold turkey without a second thought. If you don't find unique and compelling content, immersion will feel like study, like going to classes, and your willpower is not going to last long.
If there's English media that interest you, it's okay if you do English once or twice a week on schedule. If you do only Japanese, you will be constantly craving the English media you used to consume. There's a chance that you end up losing to the temptation, dropping Japanese immersion and going back to all the English content you've been abstaining from. It will be worse than occasional breaks.