"I need to" vs "I will need to"

If you say "I will need to take a shower", you mean "at some (understood) future time", which doesn't sound to me as if it matches your case, because the clause is supposed to be explaining why you can't come right now.

I think you can concoct a scenario in which it would make sense ("I can't come right now, I will need to take a shower later so I have to do my other chores now"), but without some such it will sound odd.

The fact that your shower might not be this very minute doesn't make any difference - the present tense is often used with future meaning in English (just as often as the so-called "future tense").

"I can't come right now, I need to take a shower."

This means that because you have something else to do, namely taking a shower, you can't come right now. Note that it implies that even if you aren't going, you still need to take a shower.

"I can't come right now, I will need to take a shower."

This means that you don't necessarily need to take a shower now. However, if you were to come, you couldn't do it now because you would have to take a shower first. This does not imply that you need to take a shower unless you are going to come.

"I need" is a present need. "I will need" is a conditional need at an unspecified future time.

I need to take a shower is the correct variant in the situation you describe. The present tense is used because the need exists in the present.

In the following examples the future tense is used because the need will not exist until some time in the future:

• If I decide to go out after all, I will need to take a shower.
• As soon as I arrive I will need to find somewhere to eat.