I want to add a couple things if I can between all this bickering.
I had a Dimension 8400 (which is pretty close to the 8300) at one point and I can tell you a couple things about it from my experience.
Firstly, DELL PSUs are known to be pretty bad. DELL likes to give you a bad case as well with a top mount PSU.
The heat internal to the PSU makes it perform worse. Add to that the fact that the PSU they give you in the first place is already not very good.
It isn't at all clear to me right now that there is even a problem with the ATI video card.
Power problems can disguise themselves as problems with anything else, so your lines on the screen and stuff may not even be from a bad video card. It could just be a struggling power subsystem.
It would clearly fit with everything else.
Additionally, it is worth noting here that power problems often manifest themselves as problems with fans going crazy all the time. That sounds like it matches up pretty well with your situation too.
It could really be the video card was damaged by the power system that just can't cut it anymore, or it could have been the video card going bad for other reasons, or the video card could be perfectly good.
There is no real way to tell at this point.
Also, I want to point out that how many MBs of RAM that are physically attached to the graphics card doesn't determine its performance capabilities. It does matter in terms of how many monitors you can attach and how big of monitor screens you can attach, but the processor built into the graphics card is the true determiner for how well it performs.
A HD 4870 with 512 MBs would blow away the performance of 6x Nvidia GT 220s with 512 MBs each on them if you could somehow get them to perform in parallel. That is just because the processor on the 4870 is 6x as good as the one on the GT 220s.
The HD 4870 512 MB card will blow away a lot of single video cards with 2GBs of RAM on them that are on the market too.
Without knowing the maker/model of the video cards at your disposal, I can't really advise on how much power each one would need, or if you should use any of the 512s instead of the 1GB video cards because they have faster processors on them.
However, there is something missing here that hasn't even been mentioned yet.
My Dimension 8400 case (which I assume is the same as your 8300 case) had PSU mounts that were not universal mounts. By that I mean you can't just go to the store and pick up a PSU and put it in the PC, because it physically won't fit.
I bought a PSU without giving thought to this myself and tried to put it in only to find out that the place where the power cord was supposed to be was obstructed by metal. DELL had moved the place where the cord was supposed to go in order to make sure I couldn't buy non-DELL PSUs.
I did get around this by taking metal cutters and cutting a huge ugly chunk out of the back of my case, but I just want to warn you that you may be faced with this same problem if you want to just drop by best buy or something and get a PSU.
Additionally, 1 watt from brand A doesn't mean the same thing as 1 watt from brand B. There are few to no testing standards, so the wattages written on the box don't mean very much.
Some PSUs that say 750w blow before you pull 400w from them. My PSU that said 650w on the box can easily do 815w at room temperature.
If you are only looking at the wattages on the box, you have no idea if a given PSU of 200, 300, 400, 500, or whatever wattage will be able to supply the needed power.
You get what you pay for in PSUs. If two PSUs both say that they do 450w and one of them costs half as much as the other, it can probably only provide half the power of the more expensive one too.
I can tell you that this PSU is a really good value right now after the rebate and it can deliver the power it says it can.
http://tinyurl.com/78dmqht
That PSU can definitely run most components you might be looking at, but that doesn't take into account the impact of the bad case on its performance nor does it account for the fact that it is a universal mount PSU so it might not fit the DELL case.
Additionally, I wanted to say that DELL does yet another thing to screw people, which is why both of you are having problems with video cards. At least in the past, circa the Dimension 8300 days, DELL changed the motherboard specifications around to only give some stupidly low amount of power to the video card through the PCIE slot.
The card slots themselves usually give 75w of power to whatever is plugged into them. DELL re-engineered their motherboards to only give something like 25w of power through the card slot. That means whatever you plug into it has to be a super low power video card or it won't work.
If the video card has additional power connectors that attach directly to the PSU (most new cards do) then it can suck up most of the difference from those ports, but you take a risk of damaging the card if you use them this way, because these additional ports are only meant for X amount of power, not X + 50w because the slot isn't pulling its weight.
So that could be a problem you face if you stick with the same DELL motherboard.
The DELLs were never meant to be gaming beasts and DELL always wanted to lock people into buying parts from the DELL store at 2x or more of the cost, so its pretty tough to try to modify them or resurrect them when they start having problems.
I don't particularly agree with Jamx13's way of going about saying to trash this computer and buy a brand new one, but he does kinda have a point.
Fighting with DELL's intentions about how these things should be upgraded and used might be more costly than it is worth.
Every piece in the computer is part of one big system and every piece impacts every other piece in ways that are not easy to see if you aren't highly proficient with computers.
That is if you don't have to fight with protections the maker of the PC put in to try to lock you into upgrading through them.
You would have to fight against both of those things when you are trying to upgrade this old DELL.
If you thought you might not have to replace the PSU, be prepared to whether you like it or not.
If you thought you can get by with the same case, be prepared for it not to be as easy as you thought.
If you thought you could keep the same motherboard, be prepared for problems that you haven't forseen in that regard.
Every one of those things has the potential to add to the cost of the repairs, and that isn't all either.
You may spend a whole lot more than you intended to in order to fix this DELL so it is working like you want. Just letting you know that up front.
Maybe you really don't have to, and that would be great, but you should be prepared to before you commit any money into trying to revive it.