Similar questions have been asked on here before. I posted links to those questions below AND Don Tepper's replies. I believe his to be solid advice.
Start by determining what it's actually worth. Ask your Realtor to do a CMA (competitive market analysis). That'll take a look at similar homes that have sold (or in some cases not sold) recently...certainly within the past 6 months, ideally within the past 3 months or less.
That's the most you should pay. Not offer, but pay. What you offer probably should be less than whatever the CMA suggests.
Eight months is quite a while. However, it could mean a couple of things. On the one hand, maybe by now the seller is highly motivated and is willing to take a low offer. On the other hand, it could mean that the seller is stubborn, determined to hold out for "his" price, even if it's above the market. So: Don't worry about it being on the market for 8 months. Just determine its value. Then make an offer, recognizing that you're not going to overpay for the property.
The most you should pay for a house is the lower of: (1) the house's value as determined by a CMA--a competitive market analysis--taking into consideration the house's specific condition and needed repairs, or (2) the monthly mortgage payments you can reasonably afford.
That's the most you should pay. How much under that might you pay? That all depends. Sounds as if the seller might be motivated, if he/she is paying two mortgage payments. It's likely the seller will need at least what he/she paid for the house plus about 10% to cover real estate agent fees and various selling and closing costs.
So now you have a range: The most you should pay and the least the owner is probably willing to accept. Make an offer anywhere in between. Some agents would probably suggest an offer 5% under the adjusted CMA (assuming that's in your affordability range). Others might suggest 10% under. Others, even more.
Check with your agent. Get the CMA. Get his/her advice. Then you decide.
Comparing recent sales data to recorded appraisals means absolutely nothing. It can't indicate a trend because appraisals have little to do with a home's value. Appraisals are broad-based calculations done by taxing authorities in order to collect taxes. An appraisal can easily be 15%-20% off from a property's true value. Where I live (Fairfax County), the county itself says it considers an appraisal to be accurate if it's within about 10% of the actual value of a property. So, the county would consider a $400,000 appraisal accurate if the real value were anywhere between about $360,000 and $440,000. And there are plenty of inaccurate appraisals out there, too.
Your area, and your taxing authority, probably have similar guidelines.
Besides, as you yourself note, what you're looking at are 2009 appraisals. I don't know about your area, but in mind the appraisals come out early in the year, based on sales during the past six months. In other words, 2009 appraisals are based on sales, generally, stretching back to July of 2008. And now it's May 2010. Even if the appraisals were 100% accurate (based on July 2008 data), it's now nearly 2 years later. The data are bad. Stale, inaccurate. Worthless.
Bottom line: Do not consider appraisals when making an offer. Don't do it. Make an offer based on comps. Have your Realtor run a CMA on a property you're considering. Look at what properties have actually sold for in the past 90 or fewer days. Then pay no more than the CMA figure. Probably offer less.