Sorry this is my first post. I lurk on here a lot, but it took me almost a full year to find a good reasonably priced slim fit dress shirt.
I will first say that I'm in a bit different of situation as you. I'm about 5'10" 200 pounds but a 44/33 drop. Normal fit shirts balloon out on me.
The best shirt I have found to date for under $50 are Charles Tyrwhitts. I'm not saying for a second that there aren't better options out there but this is a hint at what I have in my closet. These are all under $50 a shirt mind you, so this advice is for people who choose not to spend their entire paycheck on their wardrobe:
I own the original, slim, and tailored fit shirts
The original fits slimmer than most other shirts I own. It's a british company so they aren't made for sloppy americans
The slim fits fit me perfect-Mind you, I have a running back style build
Tailored fit-Their thinnest cut. Very slim in the chest and waist. It's definitely not an athletic cut because it has very little taper. Just overall a slim fit shirt as the name describes.
Ralph Lauren original and custom fits-poor quality, you pay for the name. I used to be a huge Ralph Lauren fanboy back in college and still own probably 50 RL shirts.
Brooks brothers slim fits(they still fit me baggy)-One of their lower end shirts. Again cheap chinese made stuff and you pay for the name.
Gold Label-regular and fitted- Honest one of the better quality cheap brands I've tried although I will say I bought about 12 of them on a Dillards sale last year for $6 each.
Murano- I feel like an intern when I wear these shirts for some reason. It's just not something I'd wear to the office.
Nordstrom Brand-Not bad for $20-25 each. I wouldn't pay much more for them than that though. They've held up to 10 dry cleanings with no buttons broken off of them. The Trim fit is one of the better fitting shirts for an athletic build. The tailored and original fits are baggy. The fitting is very similar to Brooks Brothers in my experience.
I have a few others, but like I said, hands down Tyrwhitt shirts are by far the best for the value. I got an order in about 2 months ago and ever since then 3 guys in my office have switched to them as well.
Disclaimer-No I have no affiliation to any of these companies and I'm not a style expert. I'm in a dress shirt 7 days a week at times so value is important for me.