It's really quite exhausting for me, both physically and mentally.
I'm pretty overwhelmed, mentally-speaking.
"Exhausting"とは「へとへと」や「疲れ果てている」という意味です。「そうとうきつい」の「そうとう」は"quite"(かなり)という単語を使いましたが、"pretty"も使えます。("It's really pretty exhausting for me..."
"Overwhelmed"とは「圧倒させる」や「打ちのめされる」と訳せるので、かなり深刻な状態を表す単語です。
お忙しい状況を英語で言うとしたら、このような文章が考えられます:
"Neither my parents nor my relatives live nearby. My husband comes home late, and I have a small child at home who requires a lot of attention. I am doing my best with raising my child, doing house chores, and so on. Of course, my husband is doing his best at work.
ご参考になると幸いです。