According to DampRid's site, it's made of calcium chloride crystals. Calcium chloride (CaCl2) is a ice-melting compound used in stead of rock salt (sodium chloride: NaCl). As a desiccant, it absorbs water and releases heat (in an exothermic reaction: CaCl2 + 2 H2O → CaCl2·2H2O), and you end up with a brine solution you can pour down the sink or toilet. The release of heat is what melts ice, and in solution the freezing point can be as low as -62F (-52C). This prevents refreezing depending on how much CaCl2 is in solution.

Perhaps buy some Calcium Chloride and try that.

Curiously, elsewhere in DampRid's site, it says "Damp Rid Freshner Packet" is made of:
Clay (Bentonite) 85-90%
Silica gel (amorphous) 5%
Polyethylene fiber 5-10%

This sure sounds like cat litter (clay and Silica gel). Perhaps it's cheaper (and more readily available) to buy cat litter that has the "urine-soaking" crystals (that's the silica gel or sodium silicate).

In either case, I wonder if an electric dehumidifier would be better, although it is noisier and uses more electricity.