As per the answer in the linked question, adding position = position_dodge(width=0.9) to the geom_text call lines up the values:
ggplot(meltedLikert[meltedLikert$variable != "age",], 
       aes(variable, count, fill=score)) + 
  geom_bar(position="dodge", stat="identity") +
  geom_text(data=data.frame(meltedLikert), 
            aes(variable, count, group=score, label=meltedLikert$score), 
            position = position_dodge(width=0.9),
            size=4) +
  facet_grid(age ~ .)

However, I also wanted to point out a few other things. You should no use meltedLikert$score in the aes() call; you should only refer to things in the data frame that is passed as data. Also, meltedLikert is already a data.frame, so calling data.frame() on it is not necessary (though doesn't do any harm).
The real improvement is in how you create your tabulation to begin with. Consider this instead:
tabulatedLikert <- ldply(likert[-1], function(sc) {
  as.data.frame(table(age = likert$age, score = sc))
})
ggplot(tabulatedLikert, aes(x=.id, y=Freq, fill=score)) +
  geom_bar(position="dodge", stat="identity") +
  geom_text(aes(label=score), position=position_dodge(width=0.9), size=4) +
  facet_grid(age ~ .)

You can fix the ordering of the bars by fixing them in the original data:
likert2 <- mutate(likert,
                  score1 = factor(score1, levels=c("unlikely", "likely", "very likely")),
                  score2 = factor(score2, levels=c("disagree", "neutral", "agree")),
                  score3 = factor(score3, levels=c("no", "maybe", "yes")))
tabulatedLikert2 <- ldply(likert2[-1], function(sc) {
  as.data.frame(table(age = likert2$age, score = sc))
})
ggplot(tabulatedLikert2, aes(x=.id, y=Freq, fill=score)) +
  geom_bar(position="dodge", stat="identity") +
  geom_text(aes(label=score), position=position_dodge(width=0.9), size=4) +
  facet_grid(age ~ .)

Of course, at this point, the colors don't actually add anything since everything is labeled directly on the graph, so I'd just get rid of them entirely.
ggplot(tabulatedLikert2, aes(x=.id, y=Freq, group=score)) +
  geom_bar(position="dodge", stat="identity", fill="gray70") +
  geom_text(aes(label=score), position=position_dodge(width=0.9), size=4) +
  facet_grid(age ~ .)
