Can I use a semicolon when linking a sentence fragment and a question?
Are there guidelines for using semi-colons in any of these kinds of circumstances (where a sentence fragment links with a question)? Please note that the words cannot be changed and dashes are prohibited as I am transcribing a conversation.
I was trying to leave; asking them to let me go.
She kept texting me; "What's up," "Want to go for a walk in the park?"
How did you feel about that; not having to go to work?
So why did you say what you did; "No," and later "I don't know"?
So about that cake; you made it yourself?
Talk about that for a moment; your tardiness, that is.
You bought icing; chocolate, right?
Is that something you regularly do; buy chocolate icing?
No, none of those semicolons work. A semicolon is used to connect two independent clauses (i.e. things that could be a complete sentence) that are related to each other. If you have only a sentence fragment on one side, a semicolon cannot be used; normally a comma or a dash would be called for there.
In several cases, you have a topic fragment and then a specifier fragment (e.g. "You bought icing; chocolate, right?"). In those cases, a colon is correct.
A colon can also be used to introduce a list.
(It's worth bearing in mind, given your note that these are transcribed conversations, that people do not always speak grammatically or in complete sentences. However, in all of these cases, I think you can come up with something reasonable.)
I would go with the following:
I was trying to leave, asking them to let me go.
She kept texting me: "What's up," "Want to go for a walk in the park?"
How did you feel about that, not having to go to work?
So why did you say what you did: "No," and later "I don't know"?
So about that cake—you made it yourself?
Talk about that for a moment—your tardiness, that is.
You bought icing: chocolate, right?
Is that something you regularly do, buy chocolate icing?
see This "The Oatmeal" cartoon for good information about how and when to use the semicolon.