The appropriate answer here is do what you've been taught to do, especially since you're a lawyer and likely understand the concept of negligence, breach, and damages...ie. doing what you've been specifically instructed not to do.
Realistically, the chances of you injuring someone while moving them are exceptionally low. However, there's some additional considerations from a systems manager perspective in that we want to send specific patients to specific hospitals; you should not initiate self transport if you're not trained how to triage people within your areas EMS and hospital system, such as to assess the difference between a trauma code vs a trauma consult vs a Fastrack/triage....or especially if you lack the understanding of a given hospital's trauma capabilities. For instance, a community hospital really can't do much more for a trauma patient than ambulance, and when such patients show up at a community hospital they don't actually do a whole lot other than call an ambulance to take you to the appropriate hospital. In addition, the nearest trauma hospital may be on a divert status due to the sudden influx of patients, and now your patient may sit there with little attendance on the waiting room floor while they wait for a bay to open up or an ambulance/air ambulance to transport them to the appropriate facility.
The amount of people who show up at a community hospital thinking it's a trauma is pretty significant, too. I've seen this a lot in LE planning as well, where people don't understand that just because 1 hospital in the Saint Bill SWPA Health Network is a trauma doesn't mean that all Saint Bill hospitals are trauma centers.
In such cases it's very possible you're worsening the patients condition since you're delaying definitive care, not getting them there quicker. Unfortunately, how to "do this right" beyond telling you to wait for EMS isn't a question someone can give you an accurate answer to in 5 minutes, or possibly even within a 1-2 day TECC/TCCC class unless it specifically addressed as a key topic (which I can almost guarantee it won't simply due to liability). While we default to transporting ourselves in LE instead of waiting for an ambulance, we have the advantage of a radio system that connects to a dispatcher that can tell us which hospital is best able to receive us.
I've been involved off and on with EMS since 2001 or so, have been a volunteer and professional, including rising to the position of a division chief at a fairly busy full time/professional agency. I would be hesitant to take the advice of someone telling you over the internet how to appropriately triage and move patients...like I said earlier, it's simply not a 5 minute answer, and not only requires hands on practice but also usually requires either experience or a real time link (such as dispatch) in the given system to make the judgement call. I totally agree with @ST911 on its value, I'm just not sure the answer on how to do it/how to learn it is straightforward in many locales if you're not in the emergency services.