I am having a site with publishing feature enabled. So "Save site as a template" feature is not available for me. Since I don't have control over the SP server farm, I am not sure about how I can create a WSP package for my SharePoint 2010 site. Is it possible to do with SP Designer 2010?
Yes it is possible to export a site as solution file using SharePoint designer.
Here's the tutorial you are looking for that explains step by step guide for exporting site as a Solution file using SharePoint designer,