Dzisiaj muszę odpuścić.

They both mean today.

• dzisiaj is the fuller form
• dziś is the shorter form

In normal speech, both are very common and natural. In this sentence, you could say either:

• Dzisiaj muszę odpuścić.
• Dziś muszę odpuścić.

The meaning stays basically the same.

Muszę is the 1st person singular form of musieć, which means to have to / must.

So:

• muszę = I have to / I must

A few useful forms of musieć are:

• muszę = I must
• musisz = you must
• musi = he/she/it must
• musimy = we must
• musicie = you all must
• muszą = they must

Because Polish usually leaves out subject pronouns when the verb already makes the subject clear.

• muszę already tells you the subject is I

So Dzisiaj muszę odpuścić is completely natural.

You can add ja if you want extra emphasis or contrast:

• Ja dzisiaj muszę odpuścić.
  This sounds more like I, personally, need to let it go today or I need to be the one to back off today.

Because after musieć, Polish normally uses the infinitive of the second verb.

This works like English I have to do, I must go, I need to stop.

Examples:

• Muszę iść. = I have to go.
• Muszę pracować. = I have to work.
• Muszę odpuścić. = I have to let it go / skip it / ease off.

So the structure is:

• muszę + infinitive

This is one of those Polish verbs whose exact meaning depends a lot on context.

In everyday speech, odpuścić can mean things like:

• let it go
• give up on it for now
• skip it
• back off
• stop pushing
• ease off

So in this sentence, it usually suggests that the speaker has decided not to continue, not to push themselves, or not to do something they had been considering doing.

Very often, the exact thing being dropped is understood from context.

This is an aspect question.

• odpuścić = perfective
• odpuszczać = imperfective

Here, odpuścić sounds natural because the sentence is about one complete decision for today: a single act of letting something go, skipping it, or backing off.

With dzisiaj, the perfective form fits well because it sounds like:

• Today I need to make that decision and be done with it

If you used odpuszczać, it would sound more like a repeated, ongoing, or habitual action, which is less likely here.

Often, yes — but that is completely normal.

Polish often leaves out the object when it is obvious from context. So this sentence may have an implied idea like:

• Dzisiaj muszę odpuścić trening.
• Dzisiaj muszę odpuścić pracę nad tym.
• Dzisiaj muszę odpuścić to.

But if everyone already knows what is being talked about, Polish can simply say:

• Dzisiaj muszę odpuścić.

That sounds natural in conversation.

That is a very good question, because odpuścić sobie is also very common.

There is a difference in nuance:

• odpuścić can mean dropping something, backing off, or skipping something
• odpuścić sobie often sounds more like give yourself a break, let it go, or stop demanding something of yourself

So:

• Dzisiaj muszę odpuścić.
  can suggest skipping or dropping something
• Dzisiaj muszę sobie odpuścić.
  more strongly suggests being kinder to yourself or taking it easy

Both are possible, but they are not exactly identical.

Yes. Polish word order is flexible.

The original:

• Dzisiaj muszę odpuścić.

puts dzisiaj first, which makes today the starting point or topic.

Other natural versions include:

• Muszę dzisiaj odpuścić.
• Ja dzisiaj muszę odpuścić.

The basic meaning stays the same, but the emphasis changes a little.

For example:

• Dzisiaj muszę odpuścić. = emphasis on today
• Ja dzisiaj muszę odpuścić. = emphasis on I
• Muszę dzisiaj odpuścić. = a bit more neutral

It sounds natural and everyday, and a little colloquial because odpuścić is a very conversational verb in this kind of use.

It is the kind of thing someone might say in normal speech when talking about work, exercise, stress, plans, or an argument.

In a more formal style, people might choose a more specific verb depending on the situation, such as:

• zrezygnować
• odłożyć to
• przerwać
• zaprzestać

But in ordinary spoken Polish, odpuścić is very common.

A rough pronunciation guide is:

• Dzisiaj ≈ JEE-shai
• muszę ≈ MOO-she
• odpuścić ≈ ot-POOSH-cheech

More exactly:

• Dzisiaj = ˈdʑiɕaj
• muszę = ˈmuʂɛ
• odpuścić = ɔdˈpuɕt͡ɕit͡ɕ

A few useful pronunciation notes:

• sz sounds like English sh
• ś / si is a softer sound than sz
• final ę in everyday speech is often pronounced less nasally than learners expect

Stress is usually on the second-to-last syllable:

• DZI-siaj
• MU-szę
• od-PUŚ-cić