Is this a Dell computer? I don't see that many 305W powersupplies out there. If it is, then you would have 22amps on your +12 rails and the HD 5750 should work fine if everyone on this forum claims it only uses only 66-75W.
Also, if it's a Dell, make sure you that you get a single bracket card. Double brack cards will not fit if you have a BTX motherboard, unless you go to the extreme and modify the case.
I am still running a HD 3850 on my Dell Dimension E520 for about 2 years now and it uses about 95W on my 305W powersupply.