Sentences as secondary truth bearers in a pluralistic proposition theory #2

I have had some additional thoughts about this after discussing it with fast here. First fast asks:

“You said, "a sentence is true [if and only if] it expresses exactly one proposition and that proposition is true. I don't understand the reasoning behind the "exactly one" condition as you have worded it. An implication of what you said is that a sentence that expresses more than one proposition (hence, not exactly one proposition) is not true because you said, "if and ONLY if", but I don't see why you would think that.

[...]

Is it because if one of the propositions is false, then the sentence is both true and false and that's a contradiction?”

I did reply to that in the thread but I think it deserves a longer reply. First, yes, it is to avoid conflicts with bivalence about sentences, that is, for all sentences, a sentence is either true or false but not both. But then I realized that maybe one could drop bivalence about sentences but not drop it about propositions. Supposing that one drops bivalence about sentences, then one can adopt much broader truth-conditions of sentences:

A sentence is true iff it expresses a true proposition.

A sentence is false iff it expresses a false proposition.

However it is also possible to accept broader truth-conditions even keeping bivalence about sentences. One could just specify that all the propositions expressed by a sentence has to have the particular truth value. It doesn't matter if it is one or more:

A sentence is true iff it expresses only true propositions.

A sentence is false iff it expresses only false propositions.