I work at fairly busy inner city hospital (90K annual visits, not uncommon for 50-60 in the waiting room), but no cath lab, neurosurgery, psych, peds etc or significant subspecialty back up, so transfer patients out pretty frequently. We have a specific "transfer center" that I notify and tell them this patient needs X service ASAP. They make the phone calls to the other hospitals, and call me back once it's time for a doc to doc sign out. Sure, it can sometimes be 24-48 hours before the bed becomes available and I'm signing the same patient out several days later. But at least I'm not the one making the calls and I can go and see other patients.
When you say you weren't trained for this, I think you likely mean that as a physician, your time is not best spent on the phone trying to get hospitals to accept your patient. I would argue you should talk to your admin folks and tell them that there should be someone else making those calls i.e. a secretary, RN etc...
If they don't listen to your pleas, give it a few months, someone will code in the waiting room, your ED will be on the news, and maybe things will start to magically fix themselves? Or not.
Also, we consult our hospitalist group on patients that are boarding waiting for a bed at outside facilities for some things. I could care less about sliding scale insulin dosing and DVT prophylaxis, but they leave recommendations and at least the patients do get slightly better care while waiting for a bed at the outside facility. I'm not trained in inpatient medicine, but usually its not the patients fault that there's no bed and I like the idea of having help to manage their inpatient needs. I've had some patients that were boarding so long that cards/renal gave recommendations, we diuresed the patient back to their dry weight, made some med adjustments, and discharged them after a few days of boarding in the ED. It's definitely not ideal and how things should be done, but sometimes its better than nothing.