We always leave Florida and head to the Western mountains during the Winter months. It is no problem if you don't have a set schedule and follow what is going on with the weather.
You need to check ahead with the RV parks and make sure they will have a clear space for you. ALL of the RV parks do not close but some do and you need to check ahead of time. On our trips it is not unusual for the park owner to take his front loader and clear the snow from some sites for us to park.
We spent one Christmas in the Grand Canyon Trailer Village and it was great. We were one of about 6 other campers and had deer wandering all through the RV park.
As far as water goes, we have never had a problem finding water to fill our tank. We never hook up to the park water unless it is to fill our fresh water tank. Keeping it topped off and using a 100 watt incandescent light bulb in the wet compartment has prevented any frozen water lines or pump.
The same goes with the sewer connection. We only hook up to dump. Many truck stops, like Flying J, have fresh water and dump facilities available but we have always found RV parks to take care of our needs.
We use electric heaters for most of our heat but naturally we do keep our LP tank full. Our roof top heat pump will work down to about 32 degrees and we have a portable heat pump that we have used in temps to -10. We also use electric space heaters and use an electric blanket.
We have never been uncomfortable in the cold weather and we always carry snow chains. By watching the weather I have only once had to use the chains on the road but I have used them a few times to get in to and out of RV parks.
We appreciate the lack of crowds during the Winter months at the National Parks.
We are not afraid to drive during a little snow although when the ice slides down your windshield it gets interesting. The defrosters are at the bottom so the snow accumulation at the top eventually slides down as it melts.
After many years of doing Winter travel, we have found what works and what doesn't. Our dog loves the snow and cold weather and so do we.
We have found that if we retract our slides in slowly, with someone watching outside, the snow will drop off and not cause any problems.
If we are expecting a heavy snow overnight, then we keep our slides retracted, just like they recommend during a heavy rainstorm.
Just use a little common sense and keep informed about what is going on with the weather. The last few years, the snowfall in many areas has not be that heavy. Naturally that can change at any time, which is why you need to monitor it daily.