Engelberg is only a little further away from basel that feldberg. The skiing is adequate for beginners and learners (although in my opinion not so good for intermediates, where really long blue runs & would be better. The reds & limited balcks are fine for a day out). Do remember that Engelberg, like Saas Fee, is north facing. This means you don't get to ski in the sunshine & it can be several degrees colder than other villages/resorts & inadequate kit would be a mistake!
You might like to try Davos or Klosters. They're not too far away, certainly when compared to Zermatt & french 3 valleys. You'll be lucky to find hotel accommodation at short notice though.
My best find is this website:
http://snow.myswitzerland.com/snow_reports. It is in english and has lots of well thought-out features for finding ski resorts & providing information about them.
Where ever you go, get there early (not later that 09:30) or you could end up parking a long way from the lifts, and kit hire will can take a while - Oh, and pre-book lessons!
have fun!
JC