OP, as BG said, welcome to NYC!
The majority of people have roach issues. The situation you described is rather average.
If it grosses you out to the degree it seems, then you have a problem.
There is only two ways to deal with it:
1) continue as you are. Pay heed to the comments re "water". Those things need to drink as well. Make sure the kitchen and bath are kept dry, especially at night! The kitchen sink s/b clean and dry! The ct's bowll s/n have water overnight.
2) it may be a probllem with the cat, but Lime works wonders. Place under your carpet or linoleum. Also, there is a sort of paste that comes in a tube (Ii thi k it may be similar to the stuff in roach motels or whatever)l. Place that in any 'crcks about where they make be nested. Ask the exterminator.
Maybe ask someone to keep the cat for a couple of months.,
Do the above and it should help significantly.
Other than that the only way is to earn more money in order to pay more rent. Then find a high quality well maintained upscale building. New buildings of this caliber are great, as they won't havee bugs or mice. Ask the management before rent about the issue and what measures they take. Ask tenants before renting. Also be aware of the caliber of tenants!!!
The last thing upscale people will tolerate is to be having a dinner party and a roach or mice joins the party. Managers of upscale buildings know this and provide the according quality.
Now, some people will say that income doesn't matter all buildings in NYC have a problem to one degree or another. On one hand they are right, BUT on the other they are wrong! It is all in the mainttenance of the building.
For example, my building, last Winter after the big snow storm, I was standing outside my building, I watched a large mouse climb out of the snow piled on the corner, from the sewer I believe he camme. I watched, very curious as to where he was going, and if he was headed to my building.
The mouse was freezing and apparently desparate. Snow effects everything! His natural home where ever it was must have been snowed in or something and he was seeking shelter.
First, he attempted to crash the corner trash recepticle, but was thwarted by the relative new metal cases the sanitation dept is now using, which completely covers the garbage container inside. The mouse could not find a way inside.
He gabe up on the recepticle, and then, as I feared headed to the building! I fully expected the mouse to find an opening, I actually figured he already had a door iinto the building, even though I had never seen a mouse and was unaware of any complaints.
Mine is a sizeable building, and comparatively new and modern. I watched as the mouse as the mouse started at one end of the building scooting aalong the bottom edge pressing the whole length seeking an opening. Again, he was thwarted!
The bottom edge where the building's bricks met with the sidewalk was fully sealed with no wear nor tear, and was constructed well enough to withstand the mouse's efforts to penetrate. Thwarted again, the mouse desparately headed back to the snow and toward the street. I suppose he went back to the sewer and perhaps froze to death.
I can only speculate that insects meet the same fate when attempting to penetrate the building. The difference is that insects can crwal up the side of a building, and some fly! So, in my building it appears that it is important to keep the screen door to the balconey closed when the glass door is open during Spring, Summer and warm Fall days, as I have noted insects will crawl several floors up!!
In two years, we've seen two roaches, and we have no idea where they came from. The balcony I suppose. The building is vigilant with extermination service, every month I think, and they will come the next day with just a call. Tenants have the option of allowing the exterminator in or not, by signing a list. In our sizeable building less than five names are ever on the list. We did not request service. After seeing each roach, we waited for another showing which did not come til a year later. None have been seen since.
I won't deal with these things at the rent I pay for a 2 bdrm. We plan on moving in a year or so to a 3 bdrm, we might buy. Rent will likely be double, and most in the forum will find it outrageous, but you get what you pay for. Besides its only money!
Best advice, earn more! Quickly!
I'd say $2000 is the mnimum necessary for a 1 or 2 bdrm, i order to live roach free. Plus a relatively new building with good management and good tenants. Watch out for building which are just bid 'Dorms' filllec with the post college crowd! They don't know how to live; and, excreted alcohol laced vomit is a sure roach magnet!!
PS, don't forget to line the walls with Lime to serve as a barrier against your neighbors!