You will need to say step by step exactly what you are doing. Exporting in AIFF format should sound virtually identical to the original.
When converting from a high bit depth (such as 32-bit float) to a lower bit depth, it is (obviously) necessary to “round” values that lie in-between the bit values of the lower format. Simple mathematical rounding (no dither) creates harmonic distortion caused by “quantize noise”. Dither provides an alternative method of rounding in which rounding is partly randomized so as to avoid this harmonic noise. In effect it is replacing one type of noise with a different type of noise. When applied to appropriately (basically, as long as the signal level is reasonably high) the dither noise will be less intrusive than the harmonic noise. It is impossible to convert from a high bit depth to a lower bit depth without creating noise of some sort.
The usual problems with dither are either, (1) the audio that is being converted is at too low a level so that the dither noise audibly diminishes the signal to noise ratio, or (2) the audio started off in an integer format at was not processed in any way, so that all sample values can be exactly represented in the same integer format, or (3) You are repeatedly importing into Audacity, working on it a bit, then exporting, then importing the new file into Audacity, working on it a bit and exporting, then importing the 3rd generation file, working on it a bit …
If your problem is due to (1), then the solution is to normalize the audio to a higher level prior to exporting.
If the problem is due to (2) then the solution is to temporarily disable dither before exporting (though it begs the question of why are you using Audacity if you are not processing the audio).
If the problem is due to (3) then the solution is to either change your work flow so that you save the project, then continue working on the same project (rather than repeatedly converting to a lower bit depth), or you export your intermediate audio files as 32-bit float (rather than repeatedly converting to a lower bit depth).
Issue (2) is being treated as a bug and should be corrected in a future version of Audacity.
issue (1) is probably the most common problem and is easily resolved, for example by normalizing to say -1 dB prior to export.
For a “normal” work flow, provided that the exported file is at a reasonable level, the dither noise should be insignificant.
You still have the original unedited files safely backed up don’t you?
Not that I’m aware of, but if you can give precise steps to reproduce the problem then I’ll look into it.