In reply to James Edwards:
Early April is probably a bit late for most Alpine cross-country trails, unless it's unusually cold and/or snowy. By its nature, this type of skiing tends to be based down in the valleys at an altitude range of about 1,000m - 1,500m. You might still be ok with places where the cross-country is on relatively high altitude plateaus, for example Alpe di Siusi / Seiser Alm in South Tyrop. The downhill thereabouts (Val Gardena) is ok for beginners to intermediates, don't know about off-piste.
Otherwise Norway or Finland as Doug suggests, cross-country is meant to be ok there into April. The only time I've been there was still lots of snow but it wasn't in great condition, thought that was meant to have been unusual. The challenge might be finding somewhere with decent downhill (significant vertical drops) and cross-country close together.