One wouldn't find many a new fugitive slaves in New York City during the Civil War. Enslaved people went North to mitigate against the risk of recapture, which ideally meant somewhere with a large black community in a free jurisdiction. New York City mostly fit the bill before the war, though commercial ties with the South made it consistently the most proslavery major city in the free states.

Once the war begins, going so far (or to Canada) quickly becomes unnecessary. The standard move instead is to seek out the United States military. There are some early war situations where United States officers hand over enslaved people to their enslavers when asked, but eventually the policy becomes that those enslaved by disloyal enslavers are contraband of war. Thus the military can seize them and put them to use as laborers. (Later on black men also become soldiers in the United States Colored Troops, but they're doing labor before this.) The result of that is that an abolitionist newspaper in the city remarks that all the railroads do well save the Underground, which "now does scarcely any business at all...scarcely a solitary traveler comes along." The general path of migration during the war years, and for a while after during Reconstruction, is for black Americans in the North and Canada to go back toward the South.

It's still legally possible to be a fugitive slave, with both the Fugitive Slave Act and the Fugitive Slave Clause of the Constitution still in operation, but both are effectively dead letters. In order to get a fugitive back, from NYC or elsewhere, at the very least an enslaver would have to prove his loyalty to the United States. Out in the field the military loses interest in fine distinctions like that, to the point where enslaved people from loyal states don't have much trouble convincing the officers in charge that their enslaver is one of the traitors. Showing up about does it. All of this becomes permanently moot with the repeal of the Fugitive Slave Act in 1864 and the Thirteenth Amendment at the end of 1865.

All that said, before this hiding oneself is sometimes an issue. Slave catchers aren't on constant watch at the borders of the South or in Northern cities, like the secret police of a twentieth century dictatorship. No one routinely checks new arrivals and ordinary people aren't expected to have regular forms of identification on hand. When they do show up, many places have a vigilance committee which may take note and people at risk might hide locally or skip town. It wouldn't surprise me if they tried to disguise themselves too, but I don't know of any instances of that or trying to change their accents.

But say that the human bloodhounds have arrived and gotten hold of a person. Assuming they don't just outright steal someone and get out of town in a hurry, which does happen but it's hard to say how much so because person thieves in the night don't leave good records, there would then be some kind of trial or hearing to prove that the accused really did belong to their enslaver. You need some legal process because otherwise it's even more free license for kidnappers than the system already is. This can be a proper trial, though after 1850 the Fugitive Slave Act mandates essentially a show trial hearing where only the white accuser gets to speak: "This person belongs to this other person and absconded from this place about this time" would be the general thrust. There may be documents offered to further support the position.

Defense at either sort of proceeding involves arguing mistaken identity, mostly. A white lawyer would have black witnesses who claim to have known the accused as a free man who lived in the community for some time before their alleged escape. Judges mostly seem to have listened to that testimony with both fingers in their ears and their eyes shut on the grounds that they know this is the done thing as well as everyone else does and, ultimately, most of them don't care that much. The proceedings would usually draw a large crowd of black (and some white) protesters. Sometimes that crowd turns into a rescue, which make for inspiring stories -sometimes people do get rescued- but is definitely the exception.

In the absence of active slave-catchers, there's not a lot of need to hide one's origin. Most people don't care. Blending in amounts to finding a place in the local black community, getting a job, and going about live as usual. One of the reasons a fugitive would land where they did was that such communities existed there and would help them get set up. Ideally they'd want a sympathetic white community to go along, which would kick in money to help relocate and reestablish fugitives, pay for and provide legal defense of accused runaways, and so forth. That's surprisingly easy to come by- a lot of antebellum whites really don't like the idea of forcing someone back into slavery, so fugitive assistance operations run pretty openly- but funding can be infrequent and formal organization among whites sporadic as well as prone to factional infighting over who is more correctly antislavery.

Source

Gateway to Freedom: The Hidden History of the Underground Railroad by Eric Foner, one of the few Underground Railroad books out there and the only one I know of that mainly focuses on New York City. One of Foner's students found some material in an archive and mentioned it might be interesting to him, so he made a note and checked it out the next time he was over there.