What to do about multiple sentences following a colon?

For example:

It happened only a handful of times in my lifetime: once when I went to the store. It just had to be done. There was no choice. Another when heading to the beach in 2013...

Should I just leave out the colon and use a period instead?

When a colon introduces more than one sentence, capitalize all of the sentences. For example:

To get rich quick: Invent a time machine. Use the machine for day trading. Beware of future versions of yourself (and other stalkers). Profit.

Some style guides (for example, AP but not Chicago) recommend capitalizing any complete sentence that follows a colon.

In your example, capitalizing “Once, when I went to the store” would clarify that you intend it as an (incomplete) sentence.

If the "once" is an example, it should be introduced with a comma rather than a colon. If you want to list all the cases, a colon would be legitimate, e.g., "It happened only a handful of times in my lifetime: once when I went to the store, once when I was conga dancing, and once when I was watching Plan 9 from Outer Space." "Once when I went to the store" is a subordinate clause, not a sentence, so the whole thing is one sentence.

The sentences following "to the store" aren't relevant. You aren't setting up a series of occurrences with them.