You shouldn't eliminate them. Typically the articles before rain and drizzle are necessary, because it's a specific instance of rain/drizzle that is causing your decision in the rest of the sentence.
Once the rain was down to a drizzle, I decided to move on.
The fact that you've described the rain doesn't necessarily allow you to eliminate the article. If you walked into a kitchen and described the table in great detail, it still wouldn't be grammatical to say "I sat at table." Just as the table you're sitting at is a specific table, the rain you're experiencing is a specific instance of rain.
You can eliminate the article when you're talking generally, as in
- "Rain is necessary for healthy plants."
- "I hate rain."
- "Rain turns into snow below a certain temperature."
Drizzle also needs an article in your sentence, and it should be "a" like the following example:
Once the kitten had turned into a cat, he was no longer cute.
- Using "the cat" would cause confusion, because it would imply that the specific kitten had turned into a different specific cat.
I'm adding a caveat here to say that with some particular words, and in some more poetic writing, it is sometimes alright to drop the articles.
"Spring turned into summer" is acceptable and fairly common, as is "Night became day"
If you were attempting to be poetic, you could possibly invoke those common phrases and say "Rain turned to drizzle, and I decided to move on." But with the verbs you use in your original sentence, it doesn't sound poetic--it sounds incorrect.