Your best bet IMO, is to head north and then east on US 2 to stay in East Glacier MT or head north on US 89 to stay in St. Mary or Babb. I did see a cabin to rent on the shore of Duck Lake a few miles east of Babb. The snow plowing in Glacier NP always begins on the east side of Glacier and will start soon if not already started. Try the Two Medicine area if you can't get far up the Going to the Sun Road from the town of St. Mary.
With the amount of snow that fell this winter, don't expect the Going to the Sun road to be open until late June or early July.
If you are looking for fancier accommadations check with the Meadow Lake Resort near Columbia Falls MT on the west side of the park. You can probably drive up the west side as far as the Lake McDonald Lodge.
When in Montana, you can call 511 for road conditions. the number to check on plowing in Glacier NP is 1 800 226 7623.

Your dates fall in between seasons at Yellowstone. Roads close to oversnow travel (snowmobile, snowcoach) on March 15th and don't start reopening for vehicle travel until later in spring. The only road that will be open at that time of year is the northern road between Gardiner and Cooke City which is open year round https://www.nps.gov/yell/planyourvisit/parkroads.htm There's good wildlife viewing along this route but there's no access to other parts of the park.

Access to Glacier will also be very limited at that time of year https://www.nps.gov/glac/planyourvisit/hours.htm

A few years ago my wife and I went up on the east end of the Going to the Sun Road on June 1. We made it up to a gate about 4 miles from the top of Logan Pass. They were still plowing up near the top.
The road through Two Medicine was open.
A few years before, we went up on the west side as far as Lake McDonald in November.

You probably will not be able to get close to a glacier because of the amount of snow at lower elevations. I suspect that Yellowstone, a longer drive, is a better bet for wildlife.

Not inside the park but there might be snowmobile tours in the surrounding national forest. There are rentals in Cooke City. There are very few organized activities in early April. What kind of sledding? I've gone dog sledding out of Big Sky.

If you opt for Glacier and want to see wildlife, you could try the National Bison Range on your way to Glacier but only the first 7 miles or so of the road will be open in April NBR%20Tearsheet%20Page%201%20and%202%20testfixed%20FINAL.pdf