Well, what wasn't clear from your original post is that the pdfs you were sending were already e-mail attachments. Here's an explanation (as far as I understand it).
When you open an attachment (regardless of what type) the first thing that happens is that the attachment is extracted to a temporary location on your hard drive, and it's this extracted version that's actually opened (this is why you can't for example, open an attached Word document, edit it and save it straight back into the e-mail).
It would seem that there is a "feature" in Foxit that prevents it from correctly attaching the extracted version to a new e-mail. This is common to many programs, not just Foxit.
If you were to save the attachment from the original e-mail, open it in Foxit and then send it as an e-mail, it would work fine (I'm not suggesting you do this, just trying to make clear that the problem is not with Foxit versus Adobe per se, but with Foxit and temporarily extracted files).
As the document is already attached to an e-mail, why not just forward the e-mail?