First of all, 5k is nothing when buying a house. Kiss 2500 - 8000 goodbye just at closing (depending on the loan, the offer, points, HUD, etc).
Second, you can pay stuff off and fax a copy of the statement, and perhaps a canceled check, to the credit reporting agencies and they can (and will) immediately reflect that on your report. Call first to find out the proper procedure for this. Better yet, when you work with a mortgage broker, they can usually pull the best organized, most detailed reports and collate them for you (the free ones, or even the ones you buy, are usually difficult as all hell to actually get any information out of). Brokers also work directly with the local CRAs, so you can just bring copies of statements and checks their, and they can update your numbers in real-time.
Buying a house is going to cost some coin, no two ways about it. After closing on it, every house is going to need things done (even if as simple as getting the locks changed, getting a few tools, a lawnmower, snow shovel, trash containers, new thermostat, smoke detectors, fire extinguishers, etc). If you can start off debt-free, you have a huge leg up.
Trust me, I did it the wrong way (lots of revolving debt, no cash), and I have regretted it damn near every day since. I will not buy another house with PMI. I will not buy another house that doesn't have enough equity to cover an agent's commission should I need to sell quickly (almost found myself in that position - can't afford to keep the house but can't afford to sell it, either). I won't buy a house where I can't afford to pay cash for lawn care and other tools to maintain.
If you truly have bad credit (outstanding debt has very little to do with it, until you get up into truly impressive amounts), there is a reason for it....