The bail bondsman is not the one you need to worry about here. Their role is limited to guaranteeing that you show up to court for the charges they bonded you out on. They are not law enforcement and they do not have authority over a separate warrant from a different jurisdiction.
The old DUI warrant is its own separate matter entirely. It was issued by a court in whatever jurisdiction handled that case, and that jurisdiction is responsible for serving it. When and how that happens depends on how actively they are pursuing it. Some jurisdictions pick people up quickly, others move more slowly, and some only act when you cross paths with law enforcement for another reason.
The practical reality is that with an active warrant out there alongside an existing cash bond, your exposure is real. If you get pulled over, appear in court for the theft charges, or have any contact with law enforcement, that warrant can be served on the spot. At that point, you are looking at a second bond on top of the existing one to get out on the DUI charge.
The smartest move right now is to get an attorney involved before the warrant catches up with you. In many cases a lawyer can contact the issuing court, arrange a voluntary surrender on favorable terms, and potentially get the bond set at a reasonable amount. Walking in on your own terms almost always produces a better outcome than getting picked up unexpectedly.
Do not wait for this to find you.