The .35 Whelen, wonderful round that it is, has always been plagued with this problem, ever since the first rifles were built, and the only option was opening up '06 cases.
The shoulder is small, and not all chambers are cut the same, even on factory guns. I've heard of the same problem on spankin' new 700 Remingtons, as well.
Handloading is about the only option, short of a new barrel, or setting the barrel back and cutting a new chamber.
One thing that you should know, and I've experienced it in several rifles, is that 'deep' chambers in the Whelen, or 'short' shouldered rounds, if you manage to get them to ignite, almost certainly will give cracked/separated/stressed cases after firing. What happens is that the firing pin, if long enough, will push the short round forward, giving a good amount of excess headspace. The neck will seal and grab on firing, but the back of the case will stretch to the breech face, and give separations. Bad juju.
For my loads, even with spankin' new Remington cases, I open the neck up with a .366 or .375 expander ball, then size the cases back down in the Whelen size die so that the brass will just let the bolt or action close on it. What that does is in essence, create a new shoulder that holds the round back, and lessens the chance of stretching so badly.
One other thing that I've done occasionally is to seat the bullet out to give the same effect, but you MUST use a load that is reduced, or at least just a 'starting' load from data, then neck sizing the fireformed cases and segregating them for use only in that rifle.
If you're having ignition problems in a Handi with factory rounds, the chamber or ammunition is most likely out of spec, and the only solution, if you don't or won't load your own, is to set the barrel back, rechamber, or fit a new barrel with a good chamber.
Inspect the interior and exterior of EVERY .35 Whelen case after firing for signs of stretching!