Are XAUUSD and XAUEUR correlated?

Very strongly, typically above 0.90, because XAUEUR is XAUUSD minus the euro leg and gold dominates the variance.

Can I diversify by trading different gold crosses?

No. Spreading risk across three crosses triples a single directional gold bet rather than diversifying it.

Normally XAUAUD, because the Australian dollar carries the largest volatility leg, but it is still gold.

That hedges the currency, not the gold. To reduce gold exposure, reduce the gold position.

A gold cross is gold priced in a second currency, so the return on XAUEUR is the return on XAUUSD minus the return on EURUSD.

At normal inputs every cross correlates with XAUUSD above 0.85, so holding two is one gold position at double size.