'Avon' means the same as 'River', so 'River Avon' means 'River River' or 'Avon Avon' or 'Avon River'. So I guess it doesn't really matter which you use :-)

But yes, it's normal to refer to the 'River Avon' (or just 'The Avon'). Or 'The River'...

I'll get me coat.