Which word would you use in American English instead of "top up"?

(When you put money on your cell phone)

I need you to check these sentences, too. Than you.

1) They set out on a windy day though small boats had been warned to stay off (I need a synonym) the water.
When they realized they were in trouble, they swam back to shore pulling the boat along with them.
2) However, the current pulled them (?) out to sea and their hope of escape (??) began to fade.
3) On the second day they lost their tackle, so they couldn’t catch any fish.
4) The coastguard went looking for them but couldn’t see them, though the boys waved their paddle and shouted at them
5) At last, when they had almost given up the idea of being rescued, they spotted a fishing boat. This time they were heard and they were pulled aboard the fishing boat.
6) Their rescuers asked them what they wanted to do with their fishing boat and they replied that they didn’t want to see it again

In the US, these cell phones are referred to as pay-as-you-go or prepaid cells. The terminology "top up" makes sense; my grandkids always said simply "add money" to the cell phone.

1. Change "though" to "although"
... although small boats had been warned to stay in the marina.

2. OK

3. comma after "day"

4. Coast Guard (2 words, capital letters)

5. comma after "heard"

6. comma after "boat"

In American English, the word commonly used instead of "top up" when referring to putting money on a cell phone is "recharge."

Regarding the sentences you provided, let's check them one by one:

1) They set out on a windy day despite small boats had been warned to stay off (using "despite" as a synonym) the water.
When they realized they were in trouble, they swam back to shore pulling the boat along with them.

2) However, the current dragged them further out to sea and their hope of escape started to diminish.

3) On the second day, they lost their fishing equipment, so they couldn't catch any fish.

4) The coastguard went searching for them but couldn't see them, even though the boys waved their paddle and shouted at them.

5) Finally, when they had nearly given up hope of being rescued, they spotted a fishing boat. This time they were heard, and they were pulled aboard the fishing boat.

6) Their rescuers asked them what they wanted to do with their fishing boat, and they replied that they didn't want to see it again.

Please note that I made a few adaptations to improve the overall flow and clarity of the sentences.