I do not recall your location, so this may not meet the "not too far" requirement. It will probably meet the "not too cold".
Canon City CO is called the "Banana Belt of CO" for good reason. This time of year expect highs in the low 50s. A few miles north of town on what turns into Shelf Road is mostly BLM ground. There is a BLM campground called "The Bank". Typical improved site with picnic tables, fire pit and a vault toilet. No water, no showers. It is pinyon / juniper forest surrounded by mountains.
Quite a bit of interest close by. Between Canon City and The Bank are the Cope and Marsh Dinosaur fossil quarries. You can run the Phantom Canyon > Victor & Cripple Creek > Shelf Road loop back to the Bank. I prefer running the loop starting from Phantom Canyon - this puts you on the high side on Shelf Raod.
There is a loop trail between the Bank and the near by Sand Gulch area. I only drive this one starting from Sand Gulch due to one steep and heavily eroded section with loose rock on bedrock. Fun trail anyway.
Red Canyon Park is just south of the Bank. Very nice red rock formations and driving all the way thru the park leads back into BLM land with many interesting side trails.
Further down is a BLM area called Oil Well Flats. Mountain bikers have claimed part of it but still has short 4WD trails.
Of course in Feb there can be heavy snows. https://www.allstays.com/Campgrounds-details/37371.htm
Let me know if you'd like more details.
Pikes Peak from BLM 4WD trail near Red Canyon Park
Looking west from Oil Well Flats December 2010. The Bank campground is a few miles up this valley.
From Red Canyon Park. The Bank camp is up the valley to the left.
I looked it up on the map and it falls in the 'not too far' category. Driving there is only a few hours.
We have acreage near Silver Cliff (Rosita) that we call the 'Ranch' of course it is only property with no improvements where we often go in the summer to hear nothingness. Not sure of the elevation of Banks campground but the ranch is 9700 ft. so it might be a bit snowy there.
Since I really wanted to take a bit of time and go to the ranch this (Banks) might be the perfect place. Also I was thinking of the Dunes NP but it might be not as nice as BLM land with fewer restrictions.
I'll check it out. I have a few days on my own as Deidre is going to Seattle to visit a friend and sitting home trying to go through the list of projects really isn't what I want to do this time of year.
Worst case is that it is covered with snow and the bananas are frozen.
edit: Just checked and Banks is 6800 ft.
Until recently, my brother lived north of Westcliffe. To me the Wet Mountain Valley is one of the most scenic spots in the state. But is does get cold. One of the camping websites states the Bank CG is 6,900 ft. The road leading into the camp is a little rough but I have seen RV trailers in the place.
There is another BLM CG adjacent to the Bank. It's called Sand Gulch. I do not recall whether or not it has a latrine, fire pits....etc. The road leading back to it it 100 ft south of the Bank road. The Sand gulch road is dirt and if conditions are muddy they will gate it off.
We enjoyed having some many public land options right on our door step when we lived in Canon City. Not much of that near Raton.