Can I do this by myself?
Family Law Attorney
There are a couple of things at issue here - copyright and copyright registration.
As soon as you pen your short stories, they are copyrighted. A copyright comes into being as soon as the original work becomes fixed in a tangible medium (like being stored in computer memory, being typed on a page, etc). You don't have to register your work for it to be copyrighted (nor do you have to publish to have a copyright).
While not required, it is good practice to include a copyright notice in your work (e.g. the copyright symbol, the year created and your name).
A copyright protects, within the US, your rights of publication and rights to create derivative works for the term of your life plus 70 years (there are other rules for works created before 1978). Your work may be protected under the copyright laws of other countries - there are international treaties concerning this but the specifics are far beyond the space available here.
Even though registration isn't necessary to create a copyright in your original work, you may want to register your copyright with the US Copyright office. The advantages of registration are (1) creating a public record of your copyright claim, (2) if you want to file an infringement suit later you have to register the work anyway (3) if you file the registration within five years of when the work was created, you get the advantage of "prima facie validity" (a legal - but rebuttable - presumption that your copyright is valid) (4) if you register your work within 3 months of creation, you may be eligible for statutory damages and attorney fees if you wind up suing someone for infringement (5) registration gives you the ability to record registration with the US Customs Service to prevent pirated copies of your work from entering the US from abroad.
Registration is pretty straight forward and not terribly expensive. Go to www.copyright.gov for more details.
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Please see my Legal Guide linked below.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.
Patent Application Attorney
You can either apply for copyright registration yourself with the US Copyright Office or have an attorney assist you with this.
At the present time, you may be able to register your short stories -- as unpublished works (if they are indeed unpublished) as a group and save on copyright registration fees. If you are applying for copyright registration yourself, then you can electronically file your short stories (such as PDF format).
Filing your copyright application on paper (even with the US Copyright Office's new barcoded form that you can mail in) takes quite a bit more time than filing electronically. (But, if you are registering published works, you will need to send in two copies of each work that you seek registration for.)
P.S. All of this assumes that you have not assigned/transferred your copyright to your publisher and that there is no contractual reason why you cannot claim copyright to your own works. You may need the assistance of counsel to help determine what your rights are.