It's been a while, but I though I would chime in...
I totally understand your struggles as well. It took me over 8 months in 2002 to find my first gig. And even that gig was a contracting position. There has been a lot of great advice up here that I will not repeat. I ended up doing over a year of short contract positions, before landing my first full time gig.
There are a few things that I always look for when hiring new talent, with seasoned and entry level.
First is do you have the fundamental skills? Can you sketch, think through a problem, convey an idea, etc... This sounds obvious, but what I am finding more and more is that a lot of new grads don't understand the difference between drawing and sketching. I am happy to hire someone that may not have "hot" sketching skills because they can throw down an idea and build upon it. This is super important in our field.
Second, do you have a point of view. What makes you tick? What will you bring to my team? These are 3 questions I always ask in some way shape or form in an interview. Just because you are an entry level candidate you still should have a point of view on design, you ambitions, and why you want to work for my company. Whether that is you want to influence and change peoples lives, design shoes to enhance the athlete, or elevate a brand experience through structural packaging, these are beginnings of a point of view. Take the time to sit down, craft it, and live by it. It should be reflected in your portfolio and emails you are sending to employees.
Last and kind of related to the above, is why should you be hired. Just because you want to break into ID is not a reason for me to hire you. You have to look at the company/firm you are reaching out to, understand their needs and direct you communication to those needs. What is it that you have that they cannot live without? What is it that they are doing that is super interesting to you? Why do you want to work there?
As others have mentioned, put your work up and we can give you some feedback to help you get to where you want to be.
Hope this helps.