Powinniśmy byli sprawdzić, czy w domu jest jeszcze śmietana.

It means we should have in the sense of a past obligation that was not carried out.

So:

• powinniśmy sprawdzić = we should check
• powinniśmy byli sprawdzić = we should have checked

The extra byli is what gives the sentence that past, missed-obligation meaning.

In this sentence, byli is important.

• Powinniśmy sprawdzić sounds like a present or general recommendation: we should check
• Powinniśmy byli sprawdzić means we should have checked earlier

So if you omit byli, you change the time reference and the meaning.

Because after powinniśmy byli Polish normally uses the infinitive of the action that should have happened.

So the structure is:

• powinniśmy byli + infinitive

Examples:

• Powinniśmy byli zadzwonić = We should have called
• Powinniśmy byli sprawdzić = We should have checked

Using a finite past-tense verb here would not fit this structure.

Sprawdzić is perfective, so it refers to one complete check.

That fits the idea here: they should have done a single checking action.

• sprawdzić = to check, to verify completely, once
• sprawdzać = to be checking, to check repeatedly, habitually, or over a period of time

So:

• Powinniśmy byli sprawdzić = We should have checked
• Powinniśmy byli sprawdzać would suggest something more like we should have been checking or we should have checked regularly, which is a different idea.

These forms agree with the gender of the group.

• powinniśmy byli is used for a masculine-personal or mixed group
• powinnyśmy były would be used for an all-female group

So an all-female version would be:

• Powinnyśmy były sprawdzić, czy w domu jest jeszcze śmietana.

English does not show this distinction, but Polish does.

Because Polish usually drops subject pronouns when they are already clear from the verb form.

In powinniśmy, the ending -śmy already tells you it means we.

So:

• Powinniśmy byli sprawdzić = We should have checked
• My powinniśmy byli sprawdzić is possible, but it adds emphasis, like we should have checked as opposed to someone else.

Because czy introduces a subordinate clause here, and Polish normally puts a comma before such clauses.

So the sentence is divided into:

• main clause: Powinniśmy byli sprawdzić
• subordinate clause: czy w domu jest jeszcze śmietana

This comma is standard Polish punctuation.

Because czy introduces an indirect yes/no question.

The thing being checked is basically:

• Czy w domu jest jeszcze śmietana? = Is there still any cream at home?

When that question becomes part of a bigger sentence, czy stays:

• sprawdzić, czy... = check whether / if...

So czy here means whether or if, not that.

Because Polish does not have the same strict sequence-of-tenses backshifting that English often has.

In English, after a past expression like should have checked, people often say:

• whether there was any cream left

In Polish, it is very normal to keep the present tense if the situation is viewed as current or still relevant:

• czy w domu jest jeszcze śmietana

That said, była is also possible in the right context:

• Powinniśmy byli sprawdzić, czy w domu była jeszcze śmietana.

That version sounds more clearly tied to a past moment.
The original with jest sounds natural if the question is about the state of the house/home at the time of checking, especially if that is still relevant now.

Because after w meaning in or at, the noun dom takes the locative case:

• dom → w domu

So w domu means at home or in the house.

A learner often expects something like w domie, but that is not correct. The correct locative form is domu.

Because śmietana is the subject of jest.

The clause means literally something like:

• there is still cream at home

In Polish, the thing that is there appears in the nominative:

• jest śmietana

Not accusative:

• not jest śmietanę

Also, Polish has no articles, so śmietana can mean cream, some cream, or the cream, depending on context.

One more vocabulary note: in everyday Polish, śmietana often refers to cream and very often specifically sour cream, depending on context.

Jeszcze here means still and suggests that some amount may remain.

So:

• czy w domu jest jeszcze śmietana = whether there is still any cream at home
• more literally, whether cream is still at home

With food and supplies, jeszcze often carries the idea of left over or remaining.

Yes, Polish word order is flexible, but the original order is neutral and natural.

The sentence:

• Powinniśmy byli sprawdzić, czy w domu jest jeszcze śmietana.

is a straightforward, standard way to say it.

You can move things around for emphasis, for example:

• Powinniśmy byli sprawdzić, czy jeszcze jest w domu śmietana.
• Powinniśmy byli sprawdzić, czy śmietana jest jeszcze w domu.

But those sound more marked or less natural in ordinary speech.

So for a learner, the original order is the best one to copy.