This is a little beyond the scope of Ask and Advisor, but I will try to help.
There are many potential communication channels, and you are correct to identify a plan to best get your message out. It seems that you have an immediate safety problem that needs addressing. To that end, a large sign asking residents and helps to put the carts away properly seems reasonable. It would also help if you could recruit someone to be "on call" if someone with mobility issues has difficulty storing the carts safely.
Longer term, it seems that you have to engage additional collaborators, which are the building management and residents. Together, perhaps you can decide on the best form of communication is the one that the target audience will notice, read (or watch or listen to), and engage with. What forms of communication typically work best? Do people in the building notice flyers or posters? Do you have an internal, unit-wide newsletter or forum that could be used for this purpose? Does your building use email, and will people read it? If you have a weekly unit meeting, could you use some or all of the time to make an announcement?
You might want to hold a safety training for people in the building. That way, you can ensure a captive audience. Training sessions could be face to face if there is an activity that the residents attend regularly.
It may be that different people in your unit would respond better to different forms of communication. Door-to-door vs. fliers, etc. If so, it’s ok to use more than one approach to ensure everyone receives the information.
Of course, all of the above suggestions should take available time and resources into consideration. If you are short on either, that will help you narrow down your list of possible communication plans. But getting the building or services management involved is crucial.
We hope this has been useful to you. For more information on communication planning, you might want to review some of the materials in Chapter 6 Section 1, “Developing a Plan for Communication.” If you think that your safety procedures are comprehensive enough to warrant a training, you may want to review the materials in Chapter 12 Section 2, “Designing a Training Session.”