Since it wasn't touched on in the Question and hasn't been addressed since do make sure your blade is clean. If you've been cutting a resinous wood, or certain sheet goods, the blade could have resin buildup on and around the teeth and this can significantly affect cutting performance. I wanted to lead with this as you mention you're a blade rookie so you might not be aware of this issue yet.
Assuming your saw is set up adequately there are a few things that could improve results, without having to look at a new blade as a fix. Although a purpose-made blade for the cut being done is undeniably the best thing to use for specific cuts in hardwoods. This is why they're made after all.
Do every cut twice
This sounds like a big hassle but you'd soon get used to it, and if a greatly improved cut surface is what you want and is achieved it is obviously worth it.
The first cut is slightly outside of your line, then you reposition slightly and make a second pass that basically just makes dust. Something similar is done in router work if going for a high-end result, the second pass sometimes referred to as a 'dust pass'. In addition to removing any scorching this can produce a really excellent surface, far better than you might guess.
However, some scraping or sanding should still be expected to produce the best finish possible which is my next point...
Expect to have to clean up the sawn face
For show surface it's not unrealistic to never use an edge directly from the saw. Always intend to sand, scrape or plane after sawing. This is good general practice with any cuts, as well as with all shaping done with power tools — after all, one shouldn't even use the surface left by a planer or jointer and those should be far far better than produced by even the best table saw.
Just to emphasise, this applies to show surfaces. The standards are/can be different for surfaces that won't be seen in service, and joint faces.
From the Comments:
I'm also pretty much pushing the thing as fast as I can.
That's too fast. Let the saw do the work.
This is basically a universal principle in all sawing, and it also applies to other power tools. You can get awful results with a router for example by pushing too quickly, or too high a feed rate as it would usually be said. With no changes made to the setup except for slowing down to the ideal feed rate you can get a really nice finish.
The exact speed to use will depend on species, grain and cut direction, bit type and some other factors, so some experimentation may be required.