I'm not 100% sure what you're looking for because of the wording of your question. Do you want a Kähler manifold for which every Kähler class has an extremal representative, but no constant scalar curvature representative? Or examples of manifolds with just one such Kähler class?
If it's the latter, then note that if there is a metric of constant scalar curvature in the class there can be no genuinely extremal metrics (since the Futaki invariant must vanish). (If I misunderstood your question and you already knew this then sorry for teaching you to suck eggs!) For plenty of concrete examples of extremal metrics you could look in the article of Arezzo-Pacard-Singer for blow-ups:
Alternatively, you can try the paper of Chen-Li-Sheng, which (building on work of Donaldson in the case of constant scalar curvature) settles the problem completely for toric surfaces (of which $F_1$ is an example):
From here you might be able to find more toric surfaces which have the property that all Kähler classes have extremal but not constant scalar curvature representatives. It will come down to some calculations involving polygons, but they could well be very difficult. Perhaps trying them in the case of $F_1$ would show how to find another example.