My suspicion is that a lot of the noise is coming from the camera side of things. There’s a few ways you could go about upgrading your equipment, so instead of a single answer I’ll list a couple options and explain the strengths:
If you’re looking at doing recording for production while shooting in addition to post-production stuff like foley, a portable recorder like a Zoom would be the most versatile and cost-efficient. The H1 should have a mini-jack input (please double-check! I’m on set and unfortunately can’t easily Googlr everything atm), but the H4n and H6 have the options for XLR and 1/4" inputs, which most professional mics require. If you’re also looking at upgrading microphones, I would not recommend the H1.
If you’re looking at post recording only, a cheaper option may be a computer audio interface like an M-Box, which connects to your computer and can take XLR inputs.