great

With “hand out” you want to think about things you can give someone to take in their hand. With a vaccine, they don’t really accept it in their hands, like you would a piece of paper or a giveaway. In this case, you would want to say “distribute” the vaccines, since they’re not something that someone can take in their hand. However, the main concept is right – they were then struggling to distribute vaccines fast enough.

Great example!

Couple things – we typically don’t use “woman” as an adjective. You can say, “…every female employee.” The passive voices, “is handed out” is technically correct, but sounds a little strange. In this case you might say, “Every female employee is given a gift on March 8.” However, if you want to use “hand out” you can turn it around into the active voice and say, “The company has a ritual. It hands out a gift to all female employees every March 8.”

Great, you can say “Because of the pandemic, many people hand out food for people who need it.”