You won't seem like you're "trying too hard." Don't worry about that. If your portfolio torpedoes your admission chances, it will probably be for other reasons. ;) How many pieces will there be altogether? Unless you're applying specifically to a writing major or program, it's unlikely that admission committees will do more than skim several of your submissions. Of course, if your writing is really engaging, it's certainly possible that the admission folks might get "hooked" and read everything.
The biggest danger I've seen when students send a lot of writing samples is that, typically, a couple of them may be strong but the rest are weaker. This detracts from the overall collection. So if you plan to send more than one, two, or--at the very most--three unsolicited samples, ask those teachers who are encouraging you to tell you if they feel there are some "weak links" in the portfolio. If yes, I suggest that you not submit those, even if it means that some "categories" won't be represented.
Also, make sure that your writing is on standard 8.5 x 11-inch paper with no elaborate bindings, covers, etc. Admission officials will probably want to put your portfolio right into your application folder, and if it's too big or bulky, they'll not only be annoyed, but also they may stick it elsewhere--like on some out-of-the-way shelf in the bathroom where it may not get read at all.
Hope that helps. Good luck with your applications.
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