Ditto to everything above. I know being in a small apartment isn't ideal, (DH and I have spent over 5 years in the same apartment to save for our home) but if you are serious about being homebuyers, you should probably plan to renew your lease one more year, and during that year save money seriously and do research on the home buying process and lenders. If your bank or credit union offers home buying seminars, consider attending one.
I know real estate costs vary from state to state, and I live in a high-priced area of the country, but $5000 wouldn't have gotten us anywhere. We had to pay $5000.00 just in earnest money for our offer on a townhome in order to be competitive against other buyers. That's in addition to the other upfront costs you are responsible for (remember, even if the sellers contribute some closing costs, that comes at the end.)
You need to have cash on hand for: Home Inspection (radon also, if required), appraisal fee, etc.
Also, take into account moving costs (movers aren't free unless you can rally some super awesome friends), plus moving supplies (start saving boxes & bubble wrap now!)
As many others have mentioned, home repairs. For instance, after the home inspection you'll discover things that need to be fixed, some for safety reasons. Sellers won't always agree to fix these or to credit you money toward fixing these thinks. Consider the cost of having to replace outdated appliances. Does the house come with blinds? Those costs add up.
Most people I know have had at least one month rental payment overlap with one month mortgage payment. You need money for this.
If you have overlap months, do you have the money to pay for double utilities? Home utilities are going to be different, possibly much more expensive, than apartment utilities.
Monthly mortgage payments are more than just the loan. It's also the taxes, HOA fees, PMI, etc. The less you put down, the more you'll have to pay for monthly mortgage and PMI. Also, some lenders will ding you with "loan adjustment fees" if you don't have a high enough credit score & savings - in some cases this can be thousands of dollars you have to put down at closing.
What would you need to pay up front at closing for escrow?
At closing you'll also need to pay any HOA fees. (More than $300.00 in our case, for the first month HOA, transfer of HOA membership, and document disclosure)
Prior to closing you'll also need to make arrangements for, and in some cases pay out of your own pocket, first year home owners warranty and home owner's insurance.
I don't mean to be discouraging, but buying a home can get expensive before you ever make your first mortgage payment. Buying at the top of your budget is a pretty risky situation as well, especially if you get into a bidding war.
It's easy to fall in love with houses, but I promise, there will be more you will fall in love with. When you first start out, it's easy to be starstruck by "the perfect place" but the reality is there are a number of places that will be good for the two of you.
Even if you had all of your financial ducks in a row, there's no guarantee your offer would even be entertained much less work out to closing. Not everyone gets the first home they put an offer on. If a house is appealing to you, it is probably appealing to someone else who will make an offer, so you'll have to make concessions to be the more appealing offer. Plus, along the way there are contingencies you might need to break your contract over (failed inspection, lower appraisal, etc).
Remember, there will be other houses. It might suck to stay one more year in your apartment, but they year will fly by, it will give you an opportunity to save and start sorting stuff to donate/sell/trash that you don't want to take with you,. Years from now, when you've been in your home for years, what difference will one additional year have made? Good luck!