Yup. And Feather blades. Then Dr Thayers witch hazel for after shave. Done and done.

One of the BIGGEST steps you can take to have a comfortable shave is:
Learn the GRAIN of your whiskers, and shave with the grain at first to minimize irritation.
e.g. - most of us saw our fathers shave from the cheeks down all the way past the adams apple or side of the neck, and then wonder why we got razor bumps on our necks.  Your hair likely grows in an upward fashion on your neck up to near your jawline (like upward growth of chest hair).
To map your grain - look it up on google or run your fingers in various directions on different parts of your stubble to note where the greatest resistance is...that's against the grain obviously.

Yes, razors have different blade gaps which makes them more or less 'aggressive' - a more aggressive razor can give a tighter shave with fewer passes, but the tradeoff is nicks if you don't get the angle right and can cause more irritation than a less aggressive razor.

Razors:
I prefer the Gillette Slim - 50 years old, found on Ebay for under your price range, and it's adjustable.
Adjustable means one can dial a setting which changes the blade angle from very mild to very aggressive.  Mild = more passes needed to get a baby smooth shave, yet less chance for nicks and irriation - Aggressive = closer shave in less passes.
Also have a Parker Variant adjustable someone noted above as a better option than the slim.  I disagree - I love the razor, but the head, having nice weight is bulky and makes it harder to get in tight under my nose than the slim.  Slim also beats the Gillette Fatboy in this regard, although the Fatboy is only marginally more bulky.  Merkur and Edwin Jagger are fixed angle and are good choices as well.

Blades:
Although many consider Feather sharpest, they're all plenty sharp, and I choose Astra green & white box since for me they are the right blend of sharp, yet smooth.  Astra green/white can be found on Amazon for $9/100 give or take.

Taylor of Old Bond Street Sandalwood is a favorite, but Proraso is fine and both are creamy and easy to build a protective lather.  Sterling soaps are highly reviewed as well, and come in many more scents than the two above.  Finally, there are boutique soapmakers like Reef Point Soaps, who make my very favorite 'Aviator' - heavy leather scent is like shaving in a saddle shop.

Blade gap isn't the only variable though.  The angle of the head is an oft forgotten but IMO very important part of the equation

Here's the slim vs the Fatboy:



Gillette's pre-TTO razors had a much more shallow angle.  The introduction of the TTO (first gen Superspeed and the Fatboy) led to a far steeper angle.  Evidently Gillette saw it as an issue, as the Slim adjustable and the flare tip Superspeed that came later reduced this angle significantly, but not quite to what the older razors featured.

Clausewitz8 I will keep an eye out for a Fatboy for you, still have a couple more months of flea market season to go here. You never know what you will find

Got my fatboy cheap.  I love it.