Welcome to the forum, @Dr_Batson! To export a BibTeX citation with the full journal name instead of the abbreviated journal name, go to the right-hand corner of the web app, select Settings, choose BibTeX, and you can specify that the BibTeX journal field output uses the full journal field in Paperpile, rather than the journal field (which contains the abbreviated journal names).
If you are creating a non-BibTeX citation, for example in Word or Google Docs, whether the abbreviated title or the full title is used depends on the citation style, and how the citation style uses the metadata field for each type of reference (some styles use the journal field, whereas others use the full journal field). In that case, you will need to know how the rules of the citation style you are using, and make some manual adjustments if necessary, which can be done by editing the reference in Paperpile (select the reference in the web app, and open the Edit dialog).