The PSU you have is the PC power and cooling 350 watt. Not a very good PSU, but also a bit under-rated power wise. No idea what it would deliver on the 12 volt rail which is the important number to know for running a vid card. However, some users have reported running up to 7900GTs just fine on that PSU while others have had problems. ATI 19XX series tend to draw more power then Nivida counterparts and this is true up to the 38XX series. However, ATI cards have better IQ until Nvidia released the 8XXX series. Your Dell will take a standard ATX PSU but from what I am reading some people are complaining they needed to cut a hole in the case for airflow as the case itself has a propriatory layout.
I suggest getting the card you want from a vendor with a 30 day no questions asked return policy. These types of returns do not only cover defects, but also cover people buying incompatible parts and such. Try it and if it dosen't work out, return it. Even if it does work, you may want to return it and get a much cheaper deal from an online vendor.
If you end up spending near $150, I'd spend the extra $30 for a 3850 because in the long run, your saving money with the extra life of the card and the option to one day use it in a crossfire configuration when you eventually upgrade your whole system.