When you pay, or attempt to withdraw money in the US, it will first check your USD wallet. If it’s empty, then it will automatically take money from your GBP wallet and convert it.
So no you don’t have to convert before you go.
One thing though: Revolut adds a 0.5% markup when they do this automatic conversion during the weekend, because the Foreign Exchanges are closed, so it might be better to convert beforehand. (FAQ: https://www.revolut.com/en/faq#what-exchange-rate-will-i-get )
I personally don’t bother, as 0.5% 2 days per week doesn’t usually add up to a lot, and it’s still much better than my regular bank.
The £250 limit per day for your card is temporary, it will increase over time. (FAQ: https://www.revolut.com/en/faq?lang=en#are-there-any-limits-to-topping-up-my-account )
One last thing, in case you don’t know: when you pay/withdraw cash abroad, always use the local currency (USD in the US, Euros in most of europe, etc). That way, Revolut will take care of the conversion, and you will get excellent rates.
Do do this, the merchant’s terminal will see the card is not from the US, and show you both options on screen (pay in USD, or GBP). Same thing with ATMs, it will ask you if you want their sh*t rates, or leave the conversion to be done by your own bank. You can find a couple of example here (in France, but the principle is the same): 5% ATM Fees in the First withdrawing within my country !