AirBnB is your best friend here. Expect to pay minimum US$ 1,000 for a month short-term, something reasonably small but with AC and access to a pool. Or if you want something top end, on a short-term, you'll pay up to US$ 3,500. Zona Colonial is good for tourists because you can walk around, it's probably the safest area in the country to walk around as it's heavily policed, there is a lot of nightlife and good supermarkets. Piantini and Bella Vista are similar to living in the US, lots of modern malls and condo blocks, again reasonably safe and again quite a lot to do. Your budget will stretch further if you go slightly outside these areas, so Gazcue, Ciudad Nueva and Villa Duarte all have options which are cheaper than the Colonial Zone and similarly Naco is cheaper than Piantini. But these areas are more dangerous so it depends how you live.
Remember that AirBnB charges you a 12% "service" fee which adds up quite a lot. So you may want to try to chat to the owners and find a way to do a deal direct, I've done that for long-term and it has saved me money. They block you sending numbers, but people sometimes spell their number in words (ie eight two three) and that gets through the block. Also watch out for electricity as it is extremely expensive here, so make sure it's included in the rent.
It's lowish season now so there's plenty available.