I got burned 2 weeks ago and got scorch on my hand. It hadn't been getting better until 1 week had passed
A proposed choice:
"I got burned 2 weeks ago and scorched my hand. It didn't start to get better for a week."
Why do we use this different wording? There is definitely a logical reason, however to a native speaker, one just sounds "better" than the other. To analyze a bit further: the "start" of a process is not continuous. It is more or less instantaneous. Thus, you would not use the past perfect continuous tense (had been) to describe a start.
Regarding scorch versus scorched: a Google Books Ngram search for "got scorch" shows 0 results. It's more common to use scorch in the verb form, where scorched is the past participle.