Yesterday afternoon, my younger sister told me over WeChated me that she will return to Fuyang at 5:30 pm and told me that I have no need to come to my mom's house to stay with her anymore.

She let me have a few days' rest.

She went to her daughter's work place on February 3rd of the lunar calendar.

My mom is already over ninety years old, and she needs to care for 24 hours care.

"She needs to care" would mean "she needs to be concerned [with something]."

Her daughter currently works in Hangzhou as a civil servant, and she was a new recruit inlast September last year.

The original is fine, but this sounds better to me.

So from February 3rd of the lunar calendar, I've been staying with my mom every day.

Now I can stay at my own home for several days' rest, but I don't want to do this for so long.

Just in a few days, I still want to continueI want to return to staying with my mom, in a few days; otherwise I will worry about her.

"I still want" is set in the present, not the future, which contradicts "just in a few days." It also doesn't make sense to say "continue" here because you just came from having a few days of rest.
"Otherwise I will worry about her" can stand as a sentence on its own, so it should be separated from the part before by a conjunction or punctuation such as a period, semicolon, or colon.