When we say that someone "had a full plate", it usually means that they had a lot of things to do or were very busy. There are a variety of synonyms that can be used to convey the same meaning. For instance, one might say that someone was swamped, overwhelmed, or inundated with work. Alternatively, one might say that someone had their hands full, was snowed under, or was buried in tasks. All of these phrases paint a picture of someone who is facing an immense amount of work or responsibility, and they can convey a sense of sympathy, understanding, or awe for the person who is going through it all.