For the first, it should be "Thanks for your support and kindness during this difficult time" - if you want to emphasize that it's been a difficult time for you personally, then add that at the end "Thanks for your support and kindness during this difficult time for me" For the second remove "have" and just say "I relocated to San Francisco last month" if that is the only point you're making. You could also say "In the past month, I have moved to San Francisco" but this would be used by native speakers mostly to begin a longer thought like "In have relocated to San Francisco in the past month to study English" or "In the past month I have relocated to San Francisco, started a new job, and began studying English"

You need to put a full stop/period or exclamation point at the end of your first sentence. Punctuation is required to make a sentence so a complete thought is expressed. Without the punctuation, your first sentence is not actually a complete sentence.

Darrek's answer is great. Do you mean "grammatically correct"? "Grammarly" is not an English word although it is the name of a company and their flagship product. :)

I agree that using a personal pronoun in the first sentence sounds unnatural. In the second sentence, the time phrase “last month” tells us the action is completed, so the verb should be in the simple past. You *could* rephrase the sentence as “I have relocated to San Francisco in the last month” followed by some current activity: “in order to reconnect with family”, “to begin a degree in nursing”, etc. But to use the present perfect there needs to be a connection to the present.