Well, it sounds like you have something unique to offer. Since you can differentiate yourself from the rest of the digital marketing crowd with your graphic design. Can you present a digital portfolio of illustrations, logo design, fonts and animations 2D and 3D? If so, you should have little problem landing a job in digital production. Many brands such as Starbucks, Nestle and other large corporations seek new grads who can design a new logo for its newest packaging, product line or offshoot.
Do you have such a portfolio for potential employers to view? If not, start digging in your old schoolwork. In what way have your hard skills in graphic design been tested? Where can your drafts and revisions be seen? How does it compare to all the digital design portfolios on Behance and other platforms where individuals and companies can shop for graphic designers like you?
If you have not thought about this until recent graduation, then don't panic. Find out from your classmates graduating with the same emphasis where they are finding career placement. Be pro-active. Don't just set boundaries where you aren't willing to go (unpaid work). Make a spreadsheet of people and brands you are willing to contact for an informational interview Skype or Google Hangout video. If not an informational interview, then consider initiating a meet and greet in your city/town with those who are more senior and are working professionals in graphic design careers. Choose mid and senior professionals. They are likely to be happy to provide one-time career guidance for someone like them but who is just starting out. Don't make the mistake of "luring" someone for a meet and greet coffee to "demand" a job. That's a business faux pas. Instead present yourself in your best light. Be confident, ask focused curious questions and focus less on money. In fact, never mention it in conversation. This is so you can gather information for the price of a coffee to hear who is hiring and what it takes.
Get out there and network. You will land paid work.