Not quite true. Adder. corgi and dad are all of Welsh origin but it's true that English borrows little from Welsh.English has borrowed from German, French, Norse and Latin.
But not Welsh.
When I was younger I spent a few school holidays around Merthyr where my family is from - I never heard Welsh spoken at all. But I've been back a few times recently, mainly for weddings and funerals and now hear it occasionally, so it's catching on, even in anglophone areas.
My Welsh is quite rudimentary: I can order meals, drinks, ask directions etc but would struggle with an in-depth discussion on inverted wingers and the high press.