What could be causing my sewing machine (Singer 3223) to be sewing zig zag rather than straight? Strange problem/query, but my Singer 3223 has decided to sew a small(ish) zig-zag style pattern--even when the stitch dial is set to the straight stitch. 

For clarity, the stitches in the zig zag are uniform distance from each other and consistent, they are just obviously zigzagging and not straight. I have tried adjusting tension, re-threading machine, all of that, but the problem persists.

Does anyone know what could be causing this, and if so, is it something that can be fixed?
It could be the fabric and needle match. Are you using knit fabric with a knit needle? Are you using woven fabric with a sharp needle for wovens? 

Check the manual: https://res.cloudinary.com/singer-sewing/image/upload/v1572776978/Singer-Website-Library/Manuals/Products_manuals/SINGER_3223_and_3229_Sewing_Machine_xresjx.pdf

If the issue is more like “loose stitches” the needle/fabric/thread combination can be the issue. 
Have you checked the needle size with the type of fabric? 
Otherwise, hopefully it’s still under warranty and you can get a replacement or have it adjusted.