Yeah, sorry OP, not going to get reimbursed for OAL tickets you bought yourself. This is the reason you deal with UA (in this case....but in general, whatever airline you are supposed to be on) and then let them work it out.
Truth is, rebooking on OAL, especially within 24 hours, is often not as easy as it should be, and if not done exactly correctly, leads to situations like you had. If I had to guess, agent who was on overtime, especially having mentioned it, was itching to get out and was probably not dotting her Is/crossign Ts. Within 24 hours, it's always also ideal for the agent to check with the new airline to make sure they see everything is confirmed properly. If its not have agent work with you until it is. Quite honestly, agent should know to do this, and overtime isn't really your concern - if she's not getting paid for that, that's her issue with UA, and the company should do better, but the agent shouldn't make that your problem.
So yeah, I would have looked to make sure the DL tickets were issued and good. Lesson learned (hopefully there won't be one...but for next time).
You were right to call UA back, but hanging up was not smart. Ideally, you should have looked up various options to see where tickets were available, and you can feed them to UA. As long as they are on an interline carrier, UA should be able to book you on those flights. Booking on your own was an unforced error (even if you feel UA wasn't helping you). Agents (and the computers they use) aren't always great at generically finding flights, but if you say to them, there appear to be 3 seats on, say, AA flight #### to MIA, then AA flight #### to SLC, then they can search specifically for those, and probably would find them if you could see they were bookable. Once you book on your own, UA is not going to reimburse you - any comp. is up to them, and most likely going to be just a refund of the balance of your tin not flown (probably less then you think....) and maybe some ETC/miles for your trouble. They are not going to pay a bill for last minute flights on another carrier - they have arrangements with other carriers which is why you should go back to UA to rebook - they pay behind the scenes (and not the fares you and I see).
As a last resort, someone upthread mentioned trip delay insurance - so a good idea to see if you have it and if it covers your situation - if the result of MX most should cover it. It won't actually cover new airfare - that is typically done by rebooking via the carrier that screwed up - but would cover unanticipated expense due to the delay - night at a hotel, meals, etc. So not everything, but a little is better than nothing, right?
As frustrating as the wait for an agent is, fi you don't want to pay out of pocket, unfortunately, it's just something that needs to be done. My typical playbook is to wait in line at the counter, and call the reservations number, and see which way is faster (often....it's faster to get an agent on the phone if it's the flight canceled). You had the unfortunate luck of this occurring over a busy holiday weekend - which definitely makes it more difficult, in terms of probably longer wait/hold times, and less alternate flight options with most flights full. But that is life sometimes. You just need to deal the best you can. I deal with it by assuming that a trip without a hitch is the exception, not the rule, so to expect the bumps, and be happy if they don't come. Makes it easier to deal with when they do.