WI feel stressed when I can't spend time on my own.

Sometime, I would like to be alone and free to do anything wthat I want to do.

But recently I am pressed forshort on time, so weekends are the only time that I can relax.

Pressed for is fine for meaning, this is just more natural.

When I can't spend time on my own I feel stressed.

If this was intentionally a fragment, that's okay, but I wanted to offer this option as an alternative in case it was unintentional.

Sometimes, I would like to be alone and free to do anything what I want to do.

"would" and "what" (though I think you meant "that") are both unncessary in this sentence.

When I can't spend time on my own.

Sometimes, I would like to be alone and free to do anything what I want to do..¶
"

"Sometime" refers to an unspecified point in time, whereas "sometimes" means occasionally or at times
"anything what" is wrong in this context. "Whatever" is the correct word here, as it means "anything that"

But recently, I amhave been pressed for time, so weekends are the only time that I can relax.

you're talking about present here

When I can't spend time on my own, I feel stressed.

Sometimes, I would like to be alone and free to do anything what I want to do.

But rRecently, I am pressed for time, so. As a result, weekends are the only time that I can relax.

Sometimes, I would like to be alone and free to do anything w(that) I want to do.

I put (that) in brackets because it would actually be better without the word.

But recently I am've been pressed for time, so weekends are the only time thatwhen I can relax.

After recently, you need to use the past tense.