I don’t see much of a change when viewing your screenshots on my calibrated-but-close-to-sRGB monitor directly from the forum webpage. The CMYK version is slightly duller and shifted away from yellow, but drastic is not a word I would use with the difference.
To accurately portray things in screenshots, if you haven’t, already, you probably should assign your monitor profile to the screen captures, then convert to sRGB before uploading. Another thing you might do is put both the RGB and CMYK images side-by-side and put them into one screenshot.
Basically I’m saying I don’t see a drastic change so am wondering if the screenshots I see don’t represent what you’re seeing.
Saturated Reds, Greens, and Blues are usually out of gamut in CMYK and need to be brought into the gamut printable by your chosen CMYK setup.
That appears to have happened here, where your red was not printable in your CMYK profile, and it was brought into gamut to the closest printable red.
Doing a design like that in RGB and Photoshop seems a bad idea to me.
Illustrator would, in my oppinion, be more convenient for creating vector output and converting from RGB to CMYK will likely result in 4C black which may be problematic for printing.