I know it is time consuming, but I never have more than one patient's meds on my person at a time (except in the example with the big multi drawer carts which have been phased out in a lot of facilities). It is just too easy to make a med error.
I start my shift by making sure everyone's meds are accounted for so i can request missing meds, then i gather room 1's meds take them to room 1 administer. Go back to the med room gather room 2's meds repeat...
Now this is, in part, because i came from PediLand, so room 1's Diuril and room 2's Diuril will almost never be the same dose. But even in school I got in this habit to avoid double dosing a patient or mixing up meds. I'm sure having everyone's meds on me at the same time is faster, but I've never done it that way.
If this isn't really a safety issue that applies to your unit the other thing to consider is joint commision. All sharps have to be in locked containers if in public spaces so only clinical staff have access. If there are needles in these drawers do they lock?
Medications have to be stored in locked areas where only licensed staff have access, so the drawers have to lock, and only nurses doctors or RTs should be able to get to them. If you have more than one patient's medications on you; the other patient's meds are supposed to be locked to keep from being contaminated when you are in the room.
This might be why they are making you change your practice since there is only one unlocked drawer on the new carts.
Have you considered sorting meds for each patient in the med room ahead of time, and dividing them up maybe in lab bags with the patient's sticker? That might be a little more time consuming but not much more than what you were already doing.
As per keeping meds safe. It isn't uncommon for nurses to tell families they can't get them something because they first need to safely administer medications. "I would be happy to, but first I need to give my patient their medications" People generally know that is more important. If a patient can't take a narcotic right now, depending on the hospital, I will either put it in a med cup with their sticker covering the top and secure it in the med room. Or waist it and pull another when they can.
Again the lab bags have always been my friend. I put their meds in the bag, sticker the bag, and tape it to the cart. I even tape an extra one for small trash (blister packs, alcohol swab wrappers, unit dose wrappers) you can't walk away from the meds on the cart, they need to be in your hand. But at least they are all together.