...Make a cold call and try to convince them that your material is worth reading?
Also is it ok for me to portray myself as my own manager/agent?
Don't do this. You always hear about people who manage to succeed using "stunts" of one kind or another. What you don't hear about is the overwhelmingly greater majority of people who try these kinds of stunts and get themselves permanently banned from who-knows-how many different places.
Of course you're looking to get your script read, but in the longer term you're looking to establish professional working relationships and to establish yourself as a professional writer in a community of other working professionals who know you and can trust you.
You start off doing something like pretending to be your own manager, even presuming it works for a time, for sure that's something that's going to come out, probably sooner than later and it's not going to be good news when it does.
Regarding cold calling, what I always used to do (and I have to preface this by saying that it was a long time ago so I don't know how effective it is these days) is that when you cold call and introduce yourself as a screenwriter, the moment of truth generally comes when they ask you if you have an agent.
If you just say no -- then you're dead, because they'll tell you that they don't accept unagented submissions -- good bye.
But if you say, "No, I submit my material through my attorney," -- while some places still won't go for it, for a number of places, submitting through an attorney will get you over that hump.
Now, while it would be nice to submit through an entertainment attorney, they can come steep. If you're a working screenwriter, you can establish a percentage deal. If you're not, you can't, and you'd have to pay one a retainer, presuming you could even find one.
However, if you're just looking to submit material and nothing more, any attorney with an office and some letterhead will suffice to do that. You just need a cover letter to the effect of, "As per my client's conversation with you on XYZ, I'm submitting the following screenplay, ABC, for your consideration."
If it comes down to actually negotiating a deal, of course, just any lawyer won't do -- you actually will need an entertainment attorney, and if you don't have the money to pay a retainer (which was around five grand twenty years ago) you can check into the Volunteer Counsel for the Arts, a program that most states have, which provides free or inexpensive legal assistance for members of the arts community.
NMS