NOTE: If the below solves your spamming issue please reply back and let the forum know. Perhaps if enough twitter users are made aware of the below (what access you may have given to third party sites) they’ll get mad enough to force twitter to change this and disallow third party sites from requiring that kind of access.
If you have not already done so, go to your settings and select APPS and if you have one or more apps/sites listed their with authorized access then chances are that one or more of those apps/sites is doing this. How?
When you sign into another site using your twitter account, one of the permissions you may be authorinzing is for the site to send tweets on your behalf. I have no idea why Twitter even alklows sites to require that kind of access but they do. I just find out about this myself and I immediately revoked access to all apps/sites that I had previously signed onto with mt twitter account.
FYI - A recomendation made by one developer is to have 2 twitter acounts, one you use and one you use to sign into other sites with and nothing else.