An appendix to Mere Possibilities presents a variant semantics. Stalnaker describes it thus:
It is technically feasible to do the compositional semantics in a way that mixes reference to real things with artefacts of the model and then use the equivalence relations to filter out the artifacts at the end, but one might hope to do the semantics more directly, where all the values of a complex expression...are entities that exist in the actual world. Our Tarskian semantics makes reference to infinite sequences of possible individuals, sequences that may include “merely possible individuals” that are artifacts of the model. But this was just a notational convenience. Using the methods sketched in appendix B, we could do the compositional semantics directly, where all intermediate values in the composition are actual things. (Stalnaker 2012, pp. 124–125)
However, the semantics described in appendix B addresses not the problems above, but a slightly different issue. To avoid confusion, I now explain what Stalnaker’s appendix B does and doesn’t address.
The semantic theories of appendix B and Sect. 3 differ in two relatively trivial ways.
First way. Stalnaker does not relativise satisfaction to variable assignments. Assignments are needed to handle quantification. So Stalnaker takes an alternative approach to quantification. Rather than quantifying over assignments, he quantifies over extensions of the valuation function to new constant terms. The new clause for
\(\exists\) is:

\(m, w \,\Vdash\, \exists v A\) iff, for some \(d \in D(w) : m^{[d/a]}, w \,\Vdash\, A^{[a/v]}\).
\(a\) is a new constant, not present in the original language.
\(A^{[a/v]}\) is the sentence obtained by substituting
\(a\) for all occurrences of
\(v\) bound by the initial quantifier in
\(\exists v A\).
\(m^{[d/a]}\) is a model just like
\(m\), except for its valuation
\(V^{[d/a]}\).
\(V^{[d/a]}\) is just like
\(m\)’s valuation
\(V\), except that
\(V^{[d/a]}(a)=d\). The resulting satisfactioncondition for, say,
\(\Diamond \exists x A\) involves quantification over all elements of the outer domain and consideration of all the associated extensions of the valuation function. So this is almost a notational variant of the assignmentbased approach in Sect.
3. The only real difference is that Stalnaker’s assignments are defined only for variables appearing in the sentence, rather than for all variables in the language. That could be mimicked with partially defined assignments.
Second way. Stalnaker’s valuations of constant terms are pointrelative. For a constant term
\(a\),
\(V_w(a)\) is the same object
\(d\) at every point
\(w\) whose domain contains
\(d\), and undefined otherwise. We can mimic this with pointrelative assignments subject to the same constraint. For example, the new clause for atomic predication is:

\(m, w, \alpha \,\Vdash\, \varPhi v_{1},\ldots ,v_{n}\) iff \(\langle \alpha _w(v_1), \ldots , \alpha _w(v_n) \rangle \in V(\varPhi )(w)\).
What are the gains of this approach? Think of the semantic value of an open sentence (of one free variable) as a function that takes, at a world
\(u\), each individual
\(x\) to a singular proposition about
\(x\). This function should be defined at
\(u\) only for individuals that exist at
\(u\): since there are no other individuals at
\(u\), there are no other arguments to consider at
\(u\), and no singular propositions about them to return as value. Functions defined on objects that don’t exist at
\(u\) themselves don’t exist at
\(u\). Treating assignments as insensitive to worlds forces the (formal representatives of) semantic values of open sentences to be defined relative to whatever there could possibly be (since they’re defined relative to all mere possibilia). We can avoid this by using world/pointrelativised assignments subject to the same constraint as Stalnaker places on valuations:
\(\alpha _w(v)\) is the same object
\(d\) at every point
\(w\) whose domain contains
\(d\), and undefined otherwise.
The quote above contrasts Stalnaker’s semantics with one that uses infinite sequences to determine assignments. On that approach, the semantic value of an open sentence at a world is defined relative to all such sequences. So it’s defined at some worlds relative to sequences containing things that don’t exist at that world. Interpreting this realistically will result in inaccuracy. Stalnaker’s focus on valuations for extensions of the language by a single constant, together with his worldrelativisation of valuations, avoids this (Stalnaker 2012, pp. 143–147). The semantic theory in Sect. 3 does not involve infinite sequences; it quantifies over variable assignments directly. Employing partially defined, pointrelative assignments as just described allows us to mimic Stalnaker’s approach, thereby avoiding the problems arising from satisfaction by infinite sequences.
None of this bears on the issues discussed above. The satisfactioncondition for \(\Diamond \exists x A\) still requires a witness for the satisfactioncondition of \(A\). Since its truthcondition doesn’t require a witness for the truthcondition of \(A\), the mismatch between satisfactionconditions and truthconditions remains. Stalnaker’s appendix B does not resolve and (as I read Stalnaker) is not intended to resolve, the problems with existential contingency that I have been discussing.