3 day backpacking trip with kids in April - where to go?

April can be dicey in the mountains. I've seen all kinds of weather. Even if you get a warm day the chances of 3 good days in a row are slim. I would wait till May and even then the highest elevations can get cold. If you're open to going in WV then I often tell people that the Otter Creek Wilderness is a good starter option. Go in via the Dry Fork swinging bridge and stay along the creek the entire time - just do an in and out. It's somewhat lower elevation so cold weather isn't as big of an issue and kids always like the water. There are rapids and a nice waterfall. I'm sure others on here could give you some better options if you want to be in VA.

Maybe hike or bike trip along the Cranberry River. There's quite a few shelters and that would save you from packing a big tent. Downside is that you'll have to compete with Trout fishermen for them that time of year. It would be easy walking for the kids and you could always do day hikes up the other trails.

April is unpredictable. But to be honest so is summer in the WV mountains with the localized pop up thunderstorms.

Personally, I think it wold be a great trip on bikes with kids. You could put a bike trailer on your bike for the gear.

Old Loggers Path north loop would be good, lots of gravel roads to reach it. East end of the Loyalsock even if you don't do the whole Loyalsock-Link Loop has less gravel eating. A number of the Michaux loops on the parent site could be concatenated, as mentioned. All of these are well over half an hour's drive to a reasonable size town afterwards.

One other possibility is a variant of www.midatlantichikes.com/bearmeadows-indianwells.htm (supplemental maps www.hike-mst.org/images/pdfs/SectionUpdates/Maps/2016-06-14ufisec8.pdf and www.dcnr.state.pa.us/cs/groups/public/documents/document/dcnr_20027824.pdf or the excellent Purple Lizard map of Rothrock State Forest)
I'd start from the large Forestry parking lot in Galbraith Gap - very little gravel driving off 322 compared to other options.
I'd also climb Spruce Gap trail - more interesting than Old Laurel Run Trail for kids.
Turn left to drop your pack and water in the 1st campsite past the Tom Thwaites Monument, then run around the top to the (closed) fire tower and cabin, the little vista nearby, trail register, etc.
Next day head out Mid State Trail compass-southwest, trail-north past Indian Wells, Keith Spring area. If you're really dragging, find campsites around Keith Spring. Or if much more ambitions, keep going "north" on Mid State Trail (compass-southeast) to Detweiler Junction with Standing Stone Trail, turn right briefly to a wonderful charcoal flat rhododendron-surrounded campsite just outside the Natural Area boundary.
Hit Bear Meadows on way back to car - can use the bits of the gravel Bear Meadows Road to connect back to the Jean Aron and Lonberger paths back to the car. Unless you camped around Keith Spring and shortcutting out, not much reason imo to go around the "back" side of Bear Meadows.
After the hike you are right next to State College - cheap college pizza for your crew (Home D, Hi-Way, etc.), the famous Penn State Berkey Creamery for ice cream, Growing Tree Toys downtown 1/2 block from Appalachian Outdoors outfitter in case toys are needed for little or big ones.

After doing a little more research...

Interestingly, the Blue Ridge Parkway campgrounds for the most part don't open until May while most of the Monongahela NF campgrounds are open by mid-April. So... I think we will visit Dolly Sods and Seneca Rocks which we have been meaning to do and try out the low elevation WV backpacking ideas that you gave me if the weather is warm enough.

Thanks!

Just because its south doesn't make it better for early backpacking, if that is what you are thinking. Its more about altitude. I would suggest Mike's Trout Run Circumnavigation. Nice moderate hike with plenty of views to keep the kids happy and you are not far from civilization if something happens.