Moving from MM to MC

I agree your next move should be to a good MM/MI cart. There will be a great improvement in SQ over your budget Shure cart.
Keep in mind that MC is very revealing of not only the music on the vinyl, but will reveal any flaws plus noise from your phonostage. If the cart/tonearm  is not accurately set up, the sound quality will suffer. That is why a good MC setup is expensive. You would need to spend considerably more on a phonostage than the unit you selected. The quality of the power supply is also important to achieve higher-end sonics.

I agree your next move should be to a good MM/MI cart. There will be a great improvement in SQ over your budget Shure cart.
Keep in mind that MC is very revealing of not only the music on the vinyl, but will reveal any flaws plus noise from your phonostage. If the cart/tonearm  is not accurately set up, the sound quality will suffer. That is why a good MC setup is expensive. You would need to spend considerably more on a phonostage than the unit you selected. The quality of the power supply is also important to achieve higher-end sonics.

This is what I was fearing - That it just might not be possible to get better sound than what I currently have simply by swapping out a couple relatively inexpensive parts. My hope is that I could spend ~$300 (Denon DL-103 + TC-760) to take my next big leap in sound quality, but it might not be possible at my current budget (and from what I'm hearing could actually be worse). This saddens me. 

@bstatmeister

This is what I was fearing - That it just might not be possible to get better sound than what I currently have

You will definitely get a better sound with a better MM cartridge. I have the same turntables (they are not my main turntables) and i’ve tried various MM cartridges on that stock Technics tonearm, but my arm has fluid damper and turntable was rewired, also added isonoe footers. Anyway, you don’t have to worry about it, the better cartridge has a better sound - that’s it! Pretty simple, everything starts from the cartridge. The rest you can upgrade later on step by step, but you need a decent cartridge matched to your tonearm. I told you about a guy who just swapped same Shure to a much better Ortofon M20FL Super with Nude Fine Line stylus. It was a very big improvement on the stock Technics with stock wiring etc. The cartridge is the most important thing in the analog chain in my opinion. You can change everything (cables, phono stage, speakers), but if you can’t get the information from the groove walls you will not be able to improve it by anything else. Read about different profiles of the diamond (stylus) and you will find that the best is FineLine (or Line Contact).

Conical profile of DL-103 is a nonsense, that’s the cheapest profile with rolled off bass and treble compared to FineLine or LineContact. That’s why people love to re-tip that stock Denon when they are looking for improvement. It is not the cartridge for your system, especially for your tonearm and phono stage. Forget about it! It’s a waste of money and time. It's oldschool rolled off sound for horn speakers, idler drive turntables and heavy tonearms like Thomas Schick "12.   

Wow a lot of variables to consider, turntable/arm, phono stages and the such but once you make the switch you will notice more realism to the music there is more of if thicker, darker, moodier. I switched from an ADC XLM II and a Grace F9R in 1978 to a GAS Sleeping Beauty and have never looked back, for very long, still have a SS rebuilt Grace F9R on another arm wand that I enjoy from time to time. There are folks here on Audiogon that marvel at their MM and I am sure they can make a case.

@tooblue Grace are better on light mass tonearms, the compliance of Grace cartridges are extremely high and Technics tonearm is not optimal for them, but i’ve tried my F-9F (Shibata) and F14 LC-OFC on stock technics tonearm, the F14 is much better, but even F9 is much more expensive that the OP budget as i can see.

F9-R normally goes for $600 with original stylus (cartridge body goes alone for $200 without stylus at all) and F14 for $1200 with original stylus. SoundSmith best styli for Grace looks ugly and cost $500 just for the stylus!

While the Ortofon M20FL with nude FineLine stylus cost less than $350 (it’s a bargain) and it’s a better match for Technics stock tonearm. Also original styli are still available NOS from various sources.