Learn English – Is “each” singular or plural

The meaning of the sentence changes completely if you use each vs. both:

If you say: "Both the datasets ... are stored in two schemas." then you are saying that there is a total of two schemas. (but we don't know how the datasets are divided between the two schemas (50/50? 60/40? 90/10?))

If you say: "Each of the datasets ... is stored in two schemas." then you are saying that there is a total of four schemas: two for one dataset, and two for the other.

If you want to say that there are two schemas, and that one schema contains 100% of one dataset, and the other schema contains 100% of the other dataset, then you should say: "Each of the datasets ... is stored in a schema", or "Each of the datasets ... is stored in its own schema."

Both should only be used when you want to refer to the combination (sum) of two things: The car and driver both weigh 3,510 lbs. (If they each weighed 3,510 lbs, the driver would need to go on a diet)

First, you're confusing the issue a bit with a misuse of "a record" in the adjectival sense. "There was a record of 28,000 runners participating" technically¹ means that documentation existed for this number of runners – perhaps there were more runners, but the remainder were not documented. Here, the subject is "record", so the verb is singular. If you're trying to unambiguously remark on the fact that 28,000 is a large number, you need to leave out the "of": "There were a record 28,000 runners participating." Here, the subject is "runners", so the verb is plural.

^{1 In practice, it's an ambiguous statement, because lots of people use "a record of" incorrectly.}

In the general case of "a [x] of [y]", the verb should agree with [x], unless it's one of the special cases (e.g. a lot of) where [x] itself is sorta-kinda plural in meaning, and so can take a plural verb. To figure it out, you can remove the extra words and see if the result sounds correct.

• There was a record of these statements. (There was a record.)
• The use of apples instead of pears was unexpected. (The use was unexpected.)
• It was a time of mass uprisings and protests. (It was a time.)
• There were a lot of people present. (There were a lot.)
• A variety of options were available. (A variety were available.)