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Are you talking about using an anonymous sperm donor through a sperm bank who is providing it for free or are you talking about having a friend/acquaintance giving you fresh sperm?
Under Virginia law, a donor is not the parent of a child conceived through assisted conception, unless the donor is the husband of the gestational mother. However, the definition of "assisted conception is:
a pregnancy resulting from any intervening medical technology, whether in vivo or in vitro, which completely or partially replaces sexual intercourse as the means of conception. Such intervening medical technology includes, but is not limited to, conventional medical and surgical treatment as well as noncoital reproductive technology such as artificial insemination by donor, cryopreservation of gametes and embryos, in vitro fertilization, uterine embryo lavage, embryo transfer, gamete intrafallopian tube transfer, and low tubal ovum transfer.
Therefore, artificial insemination would only prevent the donor from being the legal father of the child if the artificial insemination was accomplished by medical procedures - the turkey baster won't do.
If you were to apply for welfare, the government would demand the father's identity. Granted, sometimes that is impossible to provide (such as where there is a one night stand with a stranger, or in the case of rape or prostitution. However, if you are able to identify the donor, you must. You must also make any reasonable efforts to identify the donor. For example, if the donor, "Steve," is a friend, of a friend of a friend, you may be required to try to find Steve's name through your chain of friendship.
And termination of parental rights is not automatic in Virginia. In fact, in a case called Cartwright v. Cartwright, the Court said that a noncustodial parent may not voluntarily relinquish his parental rights at all. Parental rights of a non-custodial parent can only be terminated in the context of foster care proceedings based on excuse or neglect, or in a step parent adoption.
So, you are best off either using assistive medical technology (having a midwife provide intra uterine or intra cervical insemination) or by moving to a state where it is easier to terminate his parental rights (which is difficult in all states - some more so than others).