I had a lot of things to deal with for negotiating teams, and I had to solve many issues. / I had a lot of things to handle for negotiating teams, and I had to solve many issues.

Because my health becameis bad, I feelt more tired.

In this case, I have got to sleep well for moreto get healthier.

If not, I will be burnt out, and everything will be annoying.

I had a lot of things forto negotiatinge between teams, and I had to solve many issues.

I'm not sure what you meant here. Did you mean to say "I had to do a lot of things, like negotiating between teams, and solving many issues"?

Because my health became baworsened, I feel more tired.

If your health had already worsened before you came to work, you could say "Since my health had already worsened, I felt more tired afterwards." (I'm assuming the work is what made you tired)

In this case, I ha've got to sleep well for moreto get healthier.

If not, I will be burnt out, and everything will be annoying.

1. "burnout" is a noun, and "burnt out" is the adjective form. "burn out" is also a verb, like in "This workload is burning me out."

2. "everything will be annoying" sounds a little child-like, though it's not "wrong". You could say, "and everything will fall apart from there." (though that's a bit more extreme). Just "I'll get really annoyed" could also work.

That's something I don't want to do that.

"i don't want to do that" makes it sound like you becoming burnt out would be a choice. Although choosing to work more is a choice, most people would say getting burnt out isn't really a choice, so it sounds odd to say it like that. It's not wrong though, and I'm sure many natives might also use it.