Exactly. You must always do the basics. Separating in bags is one thing you could do in this situation, but fully, conscientiously checking and preparing one medication at a time is even better, IMO. For instance, if you had fully, carefully checked your vanco and prepared your piggyback/infusion first, it would have no longer been sitting there in the vial for you to draw into a syringe and push. And you are much less likely to hurt someone if your mistake is infusing something that should have been pushed, rather than pushing something that should have been infused. Create processes that you will rely upon every time with the first goal of eliminating risk of error itself, and then secondarily mitigating the chance of actual harm.
This is an issue in my area where occasionally both IV and IM medications are administered to the same patient. Nurses have their own methods they do every time the scenario arises, such as conscientiously checking 5Rs for the IM med and administering that first and being completely done with it before turning to the IV meds, beginning with making sure you do not have any IM-only meds left to administer. Then follow through one by one, conscientiously checking and administering each IV med.
You must decide right now, to not let others' decisions become your problem. Meaning, if a decision has been made to stretch the nursing resources thin, that does not mean that you have to hurry endlessly faster and faster. You did not decide on the staffing, but you have the responsibility to do your part to keep the patient safe regardless of the decisions other people made about staffing.
Some things will have to wait. Actual patient care and safety is to be prioritized over every other thing.
If it is impossible to accomplish this, then you must find a place to work where it is possible.