I'm going to read between the lines here and give it to you straight. You don't mention anything about what neighborhoods you have interest in. That may be because you are not too familiar with the city. Manhattan is expensive even for a small studio. On top of that you have 2 Dogs. Never mind what they weigh, you have two of them hence, the number of pet friendly apartments will dwindle.
Part of the reason you are viewing one place at a time could be a combination of what Ms. Levy said and the absolute fact that having pets will put you outside of apartments that could work but won't because of the pets. A catch-22 if you will.
Lastly, your budget totally limits any amount of apartments you could find. At $1700-$2000 you can expect to find a studio perhaps but not a two bedroom unless you go uptown or maybe Brooklyn-Queens. I love Dogs and in particular Terriers but I will tell you it will be rough going here. I certainly wish you all the best.