It sounds as though the acids and salts from your husband’s sweat are reacting with the dyes of your sheets. Rest assured the stains you see aren’t lingering dirt, it’s a chemical change. Since cranking down the AC is an expensive and wasteful solution I highly recommend switching to white sheets and spraying his side of the bedding with a vinegar / water solution before laundering. Do not use chlorine bleach, but an oxygen bleach such as Oxiclean may be a product worth trying to keep your new white sheets from yellowing.
If you aren’t ready to shell out for new sheets simply make the bed and use pillow shams. If your husband tends to be a heavy sweater, make sure you use a high quality mattress pad to protect your mattress. Good luck.
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