Can I travel within the Schengen area if I currently don't have a valid visa or residency permit?

Given other 'anomalies' in the Schengen system (eg Does tourist visa (90 days) apply after a long-term visa ends in Schengen countries? where reentry was likely to be allowed provided only that the applicant had left first - even if only minutes earlier) I very much doubt any special provision is made for the situation you describe. That just leaves the 'standard' rules where having applied for (but not yet been granted and not necessarily even about to be granted) a temporary residence permit is likely to be viewed, by a different country's representative, simply as not having a temporary residence permit. Hence no bypassing of the 90/180 day rule.

However, I agree with @phoog's Comment that you're unlikely to have problems in other Schengen countries. Mainly because the first time anyone is likely to be interested is when you leave the Schengen Area, and by that time you would hopefully have received your temporary residence permit. That should be justification enough for however long you had spent in the Schengen area, Poland or otherwise. The trick would be to avoid anything that might cause questions to be asked - at least until you are safely back in Poland.

I am stating this as per my experience with German auslanderbehörde and I suspect this may also be the same for Poland.

While your residence permit status is "in processing" and your current visa is expired, it is not possible to travel outside the country until foreign office has decision on your residence status, even within Schengen area or your home country.

Usually, in case of Germany, the foreign office provide you an "bestätigung/acknowledgment" letter stating that your residence permit is currently being processed and your previous visa status will be applicable until decision is made on your application. Note that you can't travel to other countries with this letter.

Other countries as Poland might have an equivalent of this letter, and I suspect your stamp on the passport is the same as the acknowledgement letter.

Long term visa holders as students or blue card holders can request for fiktion visa in case of emergencies, which is valid for 3 months, but issuing this visa is solely decided by the foreign office.