I've mailed I-130, AOS, AP and EAD altogether last week by Fedex to the Chicago office as instructed.
When asked on I-130 item 22 about which office to apply for adjustment of staus at the USCIS, my hubby and I decided to put down 'Hartford, CT' because we didn't want any trouble from bureaucracy and it seems to me that the district office with the jurisdiction over our residence is Hartford since we live in CT.

I am thinking about visiting the local office once I get the receipt in the hope to expedite my AP so that I can visit my illed diabetic father. The question is--which one should I go to? In terms of distance, we're closer to NY office because we are on the very south tip of CT so geographically the nearest office should be in NY. It takes only 45 minutes for me to get to NY by train, but it takes 2-3 hours to get to Hartford. On the same token, I am not familar with Hartford at all and have never been there before. As a result, I wish I can go to NY office both because of the shorter distance and my familiarity with NY. However, I am worried that the USCIS officers would only deal with cases in their own state regardless of the actual distance. Can someone please share experience with me or give some advice? Thank you very much.

Look up the District Offices on the USCIS website. It will tell you which office has jurisdiction over your place of residence.

Yes, it said 'Hartford, CT', which is in actual distance much much farther than NY from where we live. it seems to me that they go by 'STATE' not the 'ACTUAL DISTANCE'. looks like we have no choice but go to a much farther office.

You could try making an infopass appointment at the nearest office (NY) and try to get the AP there. Worst is they will tell you to go to CT - at the best you can save yourself a trip and get it right in NY - worth a shot I think. Good Luck