I mostly use semicolons when I have several simple ideas to get across and I'm trying to avoid too many short, choppy sentences.
"I went to the store; Tom said we need lettuce." This is how I would write it.
"I went to the store, Tom said we need lettuce." This is what was recommended.
There are specific rules for using semicolons, and none of them are to avoid short, choppy sentences, so your first example is wrong. The clauses are still short and choppy, so you've accomplished nothing.
There is a certain amount of latitude given to comma splices in dialogue (I don't know why), but I would never use a comma splice in my own writing. The simplest way to write that sentence is to use a period.
"I went to the store. Tom said we need lettuce."
Another way would be to insert a dialogue tag.
"I went to the store," I said. "Tom said we need lettuce."
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