You really don't have a problem here. What you need to do is simple.
But I'll explain why before
A US citizen DOES NOT NEED A VISA for tourist or business visits to Schengen so Karlson's answer is partly wrong.
You will be granted admission to the national territory of the Member State, AND to the Schengen 'area of freedom and security' simultaneously by just turning up. This is the same as a European citizen being admitted to the USA on a visa waiver, without the bureaucracy of ESTA or green W forms.
The maximum you can stay in the whole of Schengen is 90 days cumulatively in any 180 day period. So if you stay 30days in Poland, 30 days in France, 30 days in Germany, you are in some difficulty and have to leave the whole area.
Think of the Schengen Area as being the United States, and Poland as being say, Texas or California. However, Poland remains a nation state, while Texas and California stopped being countries in the 19th century. This means that it CAN have national bilateral rules. Say between Poland and Ukraine, This is potentially important which I'll get to shortly. So Karlson MAY be (partly) right as well.
The way this works is, that once you have been 90 days cumulatively in Poland, on the Schengen equivalent of a visa waiver (i.e. you just turned up and they let you in) you MUST leave the whole Schengen area and can't come back until your average time in all Schengen states has declined to less than 50% unless you get a visa (Schengen or national).
This can cause you unexpected problems. I presume what you are doing here, is an extended family or pleasure visit (since you can't work).
The information you have given may be right. But the implications of it may not be apparent, and you can fall foul of it with awkward consequences.
The Schengen Agreement contains provisions for national rules to be maintained on cross border traffic between Schengen States and third countries with which those states share a land border. (If you need a reference I could probably find authority with a bit of research.)
So its theoretically possible that you might be able to do a day trip across what in European Law is called the External Border and Poland will let you back in (remember, you don't need a visa). But, IF it decides to do so (and it may not, border guards can be arbitrary), it's doing this under National (Member State) rules, and not under Schengen rules. So guess what, even though you are in Polish territory, you may not leave Poland across the (unmanned) State Lines for the rest of your stay without the risk of, (e.g. if in Germany and stopped for a minor traffic infraction) being held, and possibly deported to the USA for exceeding your 90/180 quota.
So what you do is this.
You arrive in Poland, without a visa. Once you've settled in and got over your jet lag, you go to the Town Hall and tell them you want a temporary residence permission that exceeds the 90 days you are allowed under visa-free entry.
There'll be a bunch of form filling, checking of passports, but provided you can prove you are able to support yourself without working illegally, and you have someone with you who speaks fluent Polish (preferably a local resident of the town concerned) you will get in due course, legal temporary resident papers.
With those, you can freely travel anywhere in Schengen for the whol of your stay.
But questions like this need proper research as the rules can change.