"the rich low chords on the left might drown out the tune on the right, played on the 'thinner' higher notes. But I guess that's just because I can't play well enough yet to use the chords more sparingly and let the tune stand out."

You're right, Andy, that you can cut down the number of buttons fingered in the LH chords, or shorten their duration, and that will make the melody stand out more. Though in practice drowning out the tune doesn't seem to be much of a problem with the players I know. It's also possible to play RH harmony notes along with the melody, and thicken up the top end.

"You do not need to decide straight off. Learn single note style in different positions then combine the two. Easier said than done but achievable with practice."

I have to disagree, Dave. It makes life far easier if you're fully aware of the alternatives and have a good idea of where you're trying to get to. The lowest note on the RH of a standard C/G anglo is the B in the middle of the treble stave. An awful lot of common session tunes have a substantial part of the melody below that and, where the melody is being played by the LH, it's much harder to add chords. You simply can't produce a full harmonic accompaniment in that way. Of course if that's not what you're aiming for, it's less of an issue, and I should of course have mentioned octave style, as practiced in various forms by Robin, Will Duke, etc.

Switching from single-note to harmonic style isn't just a matter of a bit of extra practice - it's going to involve pretty much starting from scratch. Some people with plenty of patience and application do take the trouble and get good results, but they tend to say, "I wish I'd known that in the first place." I've taught anglo for about 30 years now, and adding harmony to an existing single-note style that isn't suited to it is a problem that's come up again and again.