Razor recommendations

I've been wet shaving since about August/September last year. I did wet shave with some crappy Schick cartridge thing and canned foam when I was a lot younger but I have been using a beard trimmer for over a decade to maintain stubble until now. All of this to say, I'm not a complete newbie but I'm not super experienced either.

This time around I briefly started with a Gillette Mach 3 with good results but my eyes were watering at the price of cartridges. I have since been using the Gillette King C with the included blades and some Wilkinson's sword blades. It's become a really enjoyable hobby and I've tried a few different soaps, creams and brushes. The issue is that I'm just not getting as close of a shave as I did with the Mach 3 unless I do multiple passes. When I do this however, I get pretty bad razor burn and breakouts on my neck so I do prefer single pass shaving with the grain. I've heard that a more aggressive razor could be the solution to a close, single pass shave.

My question is, where does the King C fall in the spectrum of aggressiveness and what other options would you guys recommend? I don't want anything ridiculously expensive but I'm otherwise open to any suggestions.

Cheers!