I caught a cold. The temperature rose to 37,.6 degrees Celsiums (under the arm).

There is noI don't have a runny nose, no or a headache.

Sometimes I feel tickles in themy throat.

There isI also feel pain in themy hands, but not always.

And tThe most frustrating thing is that I'm on vacation.

Are you on vacation now, or in a few days. If you're not on vacation yet, you can say "I'll soon be on vacation."

I caught a cold.The My temperature rose to 37,6 degrees Celsiums (under themy arm).

I feel weak and as if I'm slightly intoxicated.

There is noI don't have a runny nose, no or a headache.

Sometimes there is a tickles in themy throat.

There is also pain in themy hands, but not always.

It's a couple of days away.

I'm not sure what "it" is referring to here. Maybe you meant to say something like, "The end of my vacation is a couple of days away"? In cases like this, you should specify what you're talking about so that it's clear to the reader.

I caught a cold.The My temperature rose to 37,.6 degrees Celsium (s (measured under the arm).

In English you almost always use a (.) for decimals.

I feel weak and as if I am slightly intoxicated.

There isI have no runny nose, no headache.

"I have X" or "I am X", when talking about yourself.

Sometimes I have tickles in the throat.

There is alsoI also have pain in the hands, but not always.

It's a couple of days away.

This doesn't make sense here. Maybe the last sentence should be "And the most frustrating thing is that I'm going on vacation."
Other options:
"My vacation ends in a couple days"
"My cold will probably go away in a couple days"

So I'm drinking hot tea and relaxing.

The tense of this sentence needs to match the rest of the paragraph

I caught a cold.The My temperature rose to 37,.6 degrees Celsiums (under the arm).

I feel weak and as iflike I am slightly intoxicated.

This is just a stylistic thing. You could also so I feel weak and slightly intoxicated but that loses the sense of "as if"

There is no runny nose, no headache.

This is fine but more personal would be "I don't have a runny nose, nor a headache"

Sometimes I get tickles in themy throat.

There is also pain in themy hands, but not alwaysl the time.

Better would be "Sometimes I get pains in my hands but not all the time."

It's a couple of days away.

Not sure what "it" is. Do you mean the end of the holiday? You could say "I only have 2 days left of my holiday"

I caught a cold.The My temperature rose to 37,.6 degrees Celsiums (taken under the arm).

While not required per se, all the major English speaking countries use the decimal point, so it would be more natural to do so here as well.

I feel weak and as if I am slightly intoxicated.

I think it is clearer to add "I am."

There isI have no runny nose, and no headaches.

Sometimes tickles in the throatmy throat tickles.

ItThere's a couple of days awayleft until I return.

So I'm drinking hot tea and relaxing.