It would be easier for you to change your US passport with your new name , when you get your ILR visa they will go by the name that is in your current passport. Otherwise the name on your visa that is stuck in your passport and the name on your passport will not match. It is a nuisance , but best ,I think, in the long run. Also my visa wasn't due to expire for 5 years, but when I changed my name in my USA passport it was the whole 10 years again...
Another consideration is... your flight tickets have to match the name in your passport and its not nice to have to use your old name when you are traveling with your husband on holiday lol