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There a couple of causes to this type of condition.
If your vent hose is partially clogged more moisture will be retained in the dryer (it also takes longer to dry). Check that there is good air flow right out of the house. You can check by turning the dryer on and going outside to feel the air coming out of the vent. You should have a good steady flow of air there. If not, then you will need to disconnect the vent hose at the dryer and completely vaccum the hose clean. Also, if there is significant dust and lint you should vaccum the the dryer itself as best as you can. Some people will dissassemble the dryer to be sure that it is all the dust and lint are removed.
If the dryer is in the basement it will draw in air from the basement. If your basement is musty smelling then unfortunately the clothes can take on the same smell. The only fix for this is doing something to improve the basement air. A dehumidier will help solve this or improve it.
Look also for any damp or musty areas that are near the dryer. If you dry up those areas and paint them this can help.
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I read this to mean that after drying you were having this musty smell.
This can come from the washer (if you are smelling it after the wash). If you have a front load washer we do see this complaint.
You can run a cycle with 2 cups of bleach to clean out the washer to help rid this smell. There is also a new product call "fresh" that is used just for this purpose. The washer will need a cleaning once a month or so with this.
Please be sure if you do have a front load washer that you are using the new "HE" type soaps that are designed for these washers. Other soaps accomulate in the washer and cause problems (including smell).
If you would like to post your model number (from the machine inside the door) we can look at the exact machine.
Try cleaning the washer first. You can use bleach and maybe repeat once. Check clothes as they are coming out of the washer afterwards to see if the smell is there.
If not, then check around the dryer for the venting to be clear of any moist areas that might be behind the dryer that you were not aware of.
Also, if you are saying that the dryer you are using now is from the old house, then yes you can actually have the smell of the old basement in it! I have seen this happen. The organisms that may be inside the machine do not go away unless you dissassemble and clean everything.
Let me know what you find.
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