You need to decide if the house is worth $189K to you - it must be to some extent since that is what you offered. However, I understand you not wanting to pay more than it appraised for. Have your Realtor look at the comps used in the appraisal to see if they 'make sense'. If they don't and you and your Realtor think the house is really worth $189K (meaning you think the appraisal is wrong), you can pay the difference if you are able and if you really want the house. However, if you and your Realtor don't think the house is worth $189K after you have reviewed the appaisal, you can see if the seller will lower the price or meet you somewhere in the middle. Also, just because the sellers have been in the house for 35 years does not necessarily mean they are not under water - they could have refinanced and pulled cash out of the house when the prices where really high.
If you had the assistance of a Realtor, they would probably have told you the options you have if the property didn't appraise for your offered amount. I've seen buyers get 2nd mortgages or using other lines of credit to come up with the difference as well.
Today, it's quite common for sellers to reduce their contract price to appraised price. I'd be surprised that the seller had an experienced agent on their side - unless at the time, the market was volatile and changing often.