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I wouldn't recommend you apply for any new credit just prior to applying for a mortgage, the inquiry + a new account may decrease your scores initially and could take a few months to recover.
So I mainly hang out in the mortgage section, this question has been moved to the credit card section since the question is more specific to how credit cards impact credit scoring... but my layman's view of what you should do is pay down the authorized user account that has the $5k balance, as I believe that is also impacting your credit scores due to the high utilization (when we look for our clients to get added as an authorized user as a way to improve credit scores, we always recommend that they are added to accounts which have 0% or low utilization).
paging pizza1 as they might have better insight than I've provided.
Asking for a credit limit increase isn't a bad idea, or at least if there is an inquiry it should be offset if the credit limit increase is approved.