the 7750 will work 100% because it doesnt require additional 6 pin power connector for it to run. the 7770 for example requires the 6 pin power...with more power you get better speeds but you can visually inspect your power supply for 6 pin pcie cables
300W is pushing it and even then it's probably not the quality of a corsair or XFX, I would say you would be much better off buying a new PSU and it will last you much longer on the long run.
New PSU will open you up to more powerful video cards, but honestly, anything over the HD 7750 will be bottlenecked by your CPU.
I used an overclocked HD 5670 in my Dell with the stock PSU (same as yours) for a year and it was fine - silent even. The HD 7750 draws less power than the HD 5670. I'd be surprised if your system + the HD 7750 draws more than 125w at load. And Dell uses good PSUs.
the cheapest upgrade is to get the 7750 and stick with the current psu
if you upgrade the psu you can get more powerful cards but at the same time you have a dual core. if you want to upgrade the psu get a 450-500watt and get something along 7850