I would be surprised to see red thread in a zone 6 lawn in January!! Typically it is a fungal disease that affects infertile, underfertilized lawns during wet periods of the growing season....spring and fall are the typical infection periods. Even in my very mild but wet winter climate, it is unusual to see at this time of year. Most fungicides registered for this issue need much warmer temps to be effective so I'd just monitor for the time being.

And not much point in treating for moss now either :-) You will just get more before winter is over. However, you can do some research now on conditions that encourage moss development and make appropriate changes when the time comes. These conditions include one or more of the following: compacted or shallow, rocky soil, excessive shade, poor drainage/excessive moisture, low soil fertility, too low a pH and a generally sparse or unthrifty lawn.

As with almost everything horticultural, it always makes the most sense to treat and correct the underlying causes of gardening problems rather than just treating the symptoms :-)

Moss you can ignore. If it completely annoys you and/or twigs any OCD tendencies you might have, you can spray it with liquid iron (dissolve about 4 oz of ferrous sulfate from any big box store in 1 gallon of water with 1 tbsp of dish soap and spray it on the moss until lightly damp; 1 gallon goes a long, long, LONG way). That'll kill it on contact.

Red thread can manage in winter if temperatures are holding above freezing, but in MA I would think only in a very sheltered and warm location? I'd expect it to be more of a Southern thing. It's more likely to be leftovers from fall, and I'd be disinclined to do anything about it at this late date.

If you want, spray lightly with that iron solution that you have left over. What you're likely seeing is clinging spores. The iron's very low pH should take care of that.