You can trust your sources, the past would be indeed said. Etymonline says that:
The past tense form said developed from Old English segde.
That saith is present there is no doubt, being formed from the verb say and the present tense inflection for the 3rd person singular -eth. Though -eth was evolving towards -es, these two inflections coexisted for some time. Wikipedia says:
During the Early Modern period, the verb inflections became simplified as they evolved towards their modern forms: The third-person singular present lost its alternate inflections: -eth and -th became obsolete, and -s survived. (Both forms can be seen together in Shakespeare: "With her, that hateth thee and hates us all".)
Yet, in the Early modern English by Barber, Charles Laurence, it is explained that -eth was still preferred for written texts:
The King James Bible always uses -eth-, partly because of its dependence on earlier translations, but partly too, no doubt, because -eth was more formal and dignified. (p. 240)
I found interesting how,
The -eth morpheme does not usually have syncope of the vowel, but appears in its full form: giveth, walketh, etc. Exceptions are the verbs say and do, which often have the syncopated forms saith and doth, and the verb have, which has the contracted form hath. (p. 237)
The fact that both present and past tenses are used to describe the same scene in this particular scriptural passage has more to do with translation. The Greek version (Koine), which some maintain as the original version of the New Testament, also uses past and present together: ἤγειραν (Matt. 8:25 BGT) - awoke and λέγει (Matt. 8:26 BGT) - saith. I don't know about Hebrew or Latina Vulgata, but the Greek language used in the New Testament has a complicated tense and aspect system which is rather flexible.
Plus, there is a tendency in many languages to use the present tense to speak about the past. English makes no exception, according to Cambridge:
We commonly use the present simple to refer to the past when we want to make events sound as if they are happening now. We also use the present simple when we report what people say as part of a story.
PS: As an interesting detail that I will provide inspired by @Kate Bunning's comment, I will add that saith is used 1262 times in the KJV, while said, 3605 times. And this should not be surprising, since saith only designates the 3rd pers sg of the present tense, whereas said can be past tense for any person except 2nd sg (which would be saidst), but it can also be part of the present perfect: have/hast/hath said.