This is mostly expected. Your RAW conversion isn’t necessarily adding lots of new noise—you’re making existing noise more visible by brightening the image. When you increase exposure/brightness after capture, you amplify both the image signal and the noise. If you compare both files at the same brightness, the difference will likely be much smaller.

A few things are contributing:

• The camera JPEG is darker, so noise is less visible.
• Raising exposure in post increases visible noise.
• Extra contrast, local contrast, and sharpening also emphasize noise.
• You likely still have too much color/chromatic noise; that should be reduced separately from luminance noise.

To get cleaner results:

• Expose brighter in-camera when possible (without blowing highlights), then darken slightly in post rather than pushing exposure up heavily later.
• If you can add more light via shutter/aperture, lower ISO to reduce noise.
• In ACDSee, adjust luminance and chromatic/color noise reduction separately; increase chroma NR more if you see colored speckles.
• Be careful with sharpening and contrast, which can make noise look worse.

So the main issue is exposure and processing choices, not JPEG export itself.