When it comes to saying that you have no time to do something, what do you say colloquially, idiomatically, metaphorically, or formally?
I am too busy to write the letter these days as I have a load of homework assignments.
Everywhere I've been in the US, we would usually simply say "I'm too busy" or "I don't have time".
- I'm too busy to write letters, as I have a lot of homework.
- I'm too busy with homework to write letters.
- I don't have time to write letters. I have a lot of homework.
In the first example, the connection between not having time and having a lot of homework is made explicit by using 'as'. In the second, it is made clear with the prepositional phrase "with homework". In the third, it is implied but clear.
All of these work in conversational speech as well as formal and informal writing.
Anything more specific would be a personal stylistic choice by the speaker, and likely idiomatic. For example, you might say that you are 'swamped':
I'm too swamped with homework to write letters.
See a thesaurus for more synonyms of 'busy'. Most of these are less formal than the simpler versions. If you'd like help with the usage of a specific synonym, that's probably another question.
In your example, "the letter" should be "letters" because you're not talking about writing a specific letter but rather making a more general statement that you don't have time to write any letters.
Colloquially (and idiomatically), you might say "I'm so swamped with homework I don't have time to write letters."
To say it more formally, you might say "I am currently too busy to write letters because I have so much homework to do."
I'm not sure how to say it metaphorically. Perhaps others have suggestions.