回答
It would also be helpful to recognize that the essential difference between (1) Vpast + *should + Vbasic* and (2) Vpast + *should + Vpresent(aux) + Vpast-participle* relates to the respective inter-temporal relation of the two actions which are predicated of the subject.
But, first: you state you chose "should have said" because the verb ("to be worried") first predicated of the subject is in the past tense. You assumed the subsequent verb - sharing the same subject - ought agree in tense with the first verb. This is understandable.
Your assumption, however, mistakes a pragmatic question of temporal relation for a question of grammatical normativity.
BOTH of the alternatives are *grammatically* correct. But temporal context will determine which was the correct alternative in the particular case.
Consider the following inter-temporal relations:
[T = time][Tp = a past point in time][Tpp = a time continuous with or posterior to Tp][p-T = a point in time anterior to Tp][V1 = primary action of the subject][V2 = an action complement of the subject's primary action][¬ = relates to]
{(Tp) V1 ¬ (Tpp) V2} =
(1) "were worried" + "should say"
|{(Tp) V1} ¬ {(p-T) V2}| =
(2) "were worried" + "should have said"
(1) describes the past state of their being worried about what they should say at that *same past time* or in the future (from the perspective of that past time)
(2) describes the past state of their being worried about what they should have said at a time *before* that past state.
(1) and (2) are both referring to the past. But (1) describes the object of worry as a *present* action occuring *at or after that past time*. In contrast, (2) describes the object of worry as a past action which was completed *prior to* the past act of worrying.
Incidentally, note that "should have said" is not a *past tense*. It is present tense with a perfect *aspect*.
これは難しい問題ですね。
実は、「すべきである」「したほうがいい」という義務や当然、助言の意味の”shouldは、間接話法の過去形でもそのまま使われる助動詞なのです。過去形がない助動詞です。
“should have”は意味が全く違い、「すべきだったのに」「した方がよかった」と意味です。すべきだったのにしなかった、までがセットの表現です。
They were worried about what they should say.
（彼らは何を言うべきか心配していた）
They were worried about what they should have said.
（彼らは何を言うべきだったのか心配していた）
他にも、”might”、”could”、”would”、”ought to”も間接話法でも形が変わりません。
以上、ご参考になれば幸いです！