I've been using 24 exposure rolls of Fujifilm 400 speed film. I haven't really been satisfied with the quality after getting them developed at Walgreens. So, I was planning on upgrading to Fujifilm 1600, but I see it has 36 exposures per roll. Will this work well with my L35AF?
For a camera made in the 80s, it is very unlikely that you cannot use a roll of 36 exposure film.
36 exposure film merely gives you 36 pictures per roll (sometimes a few more depend on how you load it). The number of exposure itself should not affect the image quality.
welcome to the site.
Your question and your main concern appear to be different.
Your Question: Can you use 36 exposure rolls in your L35AF? - Yes, you can. The physical dimensions on the 35mm canister are the exact same for both 24 and 36 exposure film.
Your Concern: I'm not satisfied with the quality - Switching the film may or may not help - this really depends on what exactly about the quality is not good for you.
If your images are not sharp enough - you need to look into your lens and shooting technique.
If you're shooting Black and White - you need to be developing the film at home or using a photography company lab - not the drugstore. You have much more control over contrast and grain when doing the development yourself or in entrusting a good lab.
So, what about your images is dissatisfying? Also, what exact film are you using?
Any 35mm camera should accept 36-exposure rolls. If there are exceptions, they will be from the dawn of the 35mm age. The cartridge is specifically designed to take a 36-exposure roll, and cameras' take-up spools can handle it as well. (Ilford even made a thin-base film for a few years that would let you shoot 72 frames on a single roll, but it's no longer in production.)
I'd suggest posting a separate thread on your quality issues. You don't specify your technique, how old and well-stored the film was, the lighting conditions, etc., and there are many reasons why your photographs may be disappointing. Some may be the fault of the camera, some of the film, and some of you.