On the back page of this newspaper is an article that describes in some detail the eye-witness account of a *working* machine: "...the greater part of the calculating machine is already constructed....I have had the advantage of seeing it actually calculate, and of studying its construction with Mr. Babbage himself..."
Does this account describe a working computer? To help make this determination I would want to learn more about the error checking capability of this machine, which I assume was used for calculation of significant digits. A described, Babbage's calculating machine could be described as "computer-like" at least. I believe that this article reports the closest Babbage got to a actual working computer. Or just an elaborate calculator. There are no references in the article to what we today would identify as punch cards or programming.
Download the PDF and read for yourself. The article describes error checking and how results are viewed and how log tables could be calculated accurately, for use in astronomy. The article noted that regarding the printer "..less progress was made...".
In the 1830's news from Europe would take a few weeks to reach New York, as boat was the only cross-Atlantic communications. Most articles in the paper refer to end of December 1832 events, including the account of the Calculating Machine.
Overall there was a lack of contemporary articles about Babbage's calculating machine in 1832-35. I was able to find just the one often quoted article by D Lardner: "Babbage's Calculating Engines" from the Edinburgh Review 59 (1834) pg 263-327. The New York Weekly Messenger article above was printed a year prior, making it among the first known articles about Babbage's calculating machine. The apparent lack favorable press and the loss of funding in 1834 obscured the fact that the machine had a working visual "display" and was probably more complete than people realize. Was the calculating machine programmable? Maybe. Most writings about the Babbage projects were published a generation afer the project was cancelled and I don't think that it's possible to discount the notion entirely.