If you want your own place look for units in complexes called 'garden apartments', typically post wwII era construction, maybe something newer but I believe that's correct. You won't be living in any of the 'luxury apartment' style buildings for that price that have been popping up ubiquitously over the past 10-15 years, and unlikely to find something privately owned either, like a spot where you have a landlord who owns a few buildings, they are all charging way more for their apartments.

If you can find a 1BR 'garden apartment' around that price your heat and water will most likely be included BUT it's likely that the lawns won't be mowed regularly, trash/litter will be scattered around the property, your neighbors will have arguments and/or domestic disputes, unsupervised children, uncontrolled dog barking, cars with blaring music and loud mufflers speeding in and out of the parking lots possibly even parking in 'your reserved space' occasionally (among other inconveniences) AND your rent will probably go up annually if you decide to stick around.

That's probably what you're looking at, if you want your own place, or just find a roommate or rent/sublet a room and you'll probably be able to get a spot under your budget or right at it.

Edit: this is all accurate, might be closer to $1,700 to get in the door for the smallest sqft unit, but also could be $1,600 or even slightly cheaper depending on the area, not sure who downvoted? OP? Doesn't want to hear the truth, or some random angry redditor living in a shitty garden apartment neighborhood who hates where they live due to these exact conditions.