An entire apartment in Manhattan under 3.5k pm is going to be difficult. Your best bet is to look at rent stabilised ones. You’ll find a list online - cold call/email for availability.
At that budget you can get a studio easily, but I wouldn’t recommend those. Space aside, your clothes (and everything else) will smell of the food you cook.
Next estimate how much time do you have. Good leases are booked out 6 months in advance more often than not. You’ll have to be incredibly lucky to find a good price within a month, especially during winters. But you got the edge with long term leases, so that’ll get you in better doors.
Joining FB groups will be futile as most of the people will be looking for roommates - which isn’t your criteria. Still join them as it’ll give you an idea about rental costs in the area. Armed with that info, check for listings on FB marketplace. You can filter them by area. It also helps with verifying the owner as legit. Meanwhile keep looking on apartments.com (they’re like the ebay of rentals). It’s free and people who post on fb are likely to post there as well.
Speaking of FB, be wary of scams. It’s a shit show. Do not engage with new accounts. Do not agree to pay/cover any security deposit and/or application fee without talking to the owner/property manager first.
If you are subleasing, the sublessee agreement needs to come from the owner and any payment must be done with their consent. But you also need to have a separate agreement with the sublessor about deposits. This is because the property might already be damaged when you move in, and it’ll come from your deposits at the time of move out.
Lastly, unless you have a SSN, your deposits are going to skyrocket. Read through the terms of return. Look for damages on the day of move in; document even the smallest of the things and email the owner/leasing manager to leave a paper trail.
In my experience local landlords are kinder and leasing with them is much cheaper. They won’t charge you any application fees either. On the other hand large scale property managers have 100% of the time duped me of security deposits, in one form or the other. They’ll also have application fees (for background checks) but anything higher than $100 is a scam.
Crosscheck the apartment history on Zillow, it’ll enlist owner sometimes. Local landlords most often than not wait for the move in date for security deposit. Property managers would want it on the day of signing the lease. If anyone asks for the same in talking stage, it’s a scam. Always use a credit card, or cheque to pay for deposit/rents. Never pay application fees using venmo, cash app, PayPal etc. Always use the application payment portal.
That’s all I can think for now; might add more later. Feel free to shoot if you’ve any questions.