I should have been a little more precise when talking about "authorization" as what I was talking about and likely the many password requests you get is not "authorization" that takes the form of the number of computers that are authorized to play copy protected iTunes purchased content (which does not apply to any music you buy from iTunes anyway). This was at least partly because I could not recall the phrasing that Apple using when those annoying requests for the password pops up. Amazingly enough, I have not gotten one of those password requests in an usually long time. If memory does serve me (my memory is working better today), Apple terms those requests as "verification" rather than "authorization"...at least in the situations that effected me that I mentioned. Technically, you should only have to authorize a computer once (unless you revoke the authorization), but could have many requests for verification. And of course, if you some how log out of your iTunes account, then you will have to log in.
Now, with music, you should be able to play any iTunes purchased music (as long as it was purchased after about 2009 if memory serves...that is when Apple got rid of copy protection for all music...but even then, you should be able to redownload any purchased music in non-copy protected form unless the music label did not allow Apple that option) in iTunes even if your computer is not authorized and not logged into an iTunes account...I believe. So, I believe you should be able to log out of the iTunes account and then still be able to use iTunes and potentially not get nagged about passwords, etc.
As to converting the AAC files to MP3, since I don't have the types of issues with iTunes that you have had, I have just always used iTunes. It is not hard to convert in iTunes, but it is not exactly perfectly straight forward and obvious either. If you think you want to try doing it in iTunes (either after testing logging out of the iTunes account and finding that it will still play the songs or still logged in), then I can give you the steps in the process. Since I am assuming you don't want to do that, I won't bother unless you tell me you want to try. If you do it in iTunes, you can likely do your entire batch of about 150 songs in likely less than 5 minutes. I did the test song just to see how long it took and it was pretty instantaneous (i.e. 1 second or less for one song).
As for other options, I am not aware of any good quick, easy converter. And a quick search did not turn up anything quick. You can using a more general audio editing program, but that will take longer as you will have to open each file in the audio editing program, then "save as" or export that file to a MP3. On a Mac, you could use Audacity, which is a general sound/audio editing program.
I will spend some more time looking to see if I can find something that I am willing to suggest (I don't tend to suggest something unless I have tried it myself and some of the ones that popped up are suspicious looking, so I am not sure I would even want to test them). Do you have a Windows computer as well? I want to say you do, but my memory is not as good as it used to be. And you could also try posting a request for AAC to MP3 converter suggestions in the Audio and Video forum on this site or look through past posts to see if someone has asked before. Just be aware that you MIGHT get some what I consider to be potentially questionable suggestions.
As I noted, you can also just use a different player. An excellent free player (for both Windows and Mac) that should be able to play unprotected AAC files is VLC media player: http://www.videolan.org/vlc/
. And if you have an Android smartphone, there is a VLC media player for it as well that should allow you to play AAC files on your Android phone (there is one for iOS/iPhone as well, but an iPhone can natively play AAC files).
As for Amazon.com purchased digital music, it should already be in MP3 format that can then be played in any player or device that can play MP3 files. And FWIW, in case you did not know, if you have bought audio CDs from Amazon, then you likely also have the ability to download MP3 versions of those songs. Amazon offered that feature a while ago and I assume they still do (it has been a while since I bought an audio disc from them).