Is it appropriate for a person to adopt by themselves, or for a woman to be artificially inseminated, in order to have a child without having to wait for a partner to marry?
In answering your question we first must remember that Christians have liberty in all matters which are not expressly or implicitly prohibited by Scripture. In the cases of artificial insemination or adoption outside of marriage, we believe the Bible implicitly prohibits these actions in most if not all cases.
First, the Bible teaches that the act of procreation is reserved for a "husband and wife" in the context of marriage. In fact, God's natural order for Creation requires a man and woman to engage in sexual relations in order to produce a child. Since marriage is a prerequisite for sexual relations (see Genesis 2:23-24) and sex outside marriage is specifically prohibited as fornication (see Matthew 15:19), we believe the Bible implicitly prohibits any effort to produce children outside the context of marriage.
Therefore, any effort to become pregnant outside of marriage is sin, because it defies the natural order established by God and is effectively fornication even though a sexual act is not involved. A single woman engaging in artificial insemination for the purpose of producing offspring without waiting for marriage is no different than a woman seeking a lover and engaging in fornication for the same purpose. Artificial insemination (for a single woman) is fornication by other means, since she is defiantly opposing the order and will of God (i.e., not waiting for marriage before producing children).
Likewise, a widow who is chooses to be inseminated with frozen sperm from a deceased husband is acting outside the limits of Scripture, since she is no longer married, according to Scripture (see Romans 7:2), and therefore her opportunity to produce children has passed. Pursuing children after a husband has died is a sinful attempt to cross natural boundaries established by God.
On the other hand, the medical procedure itself is not sinful, therefore we see no reason a wife cannot use artificial insemination to overcome a fertility issue within the context of marriage and assuming she has the cooperation of her husband. In such a case the wife remains within the boundaries established by God, since she is obtaining medical help to procreate with her husband.
Regarding adoption outside marriage, the same prohibitions would apply. God's design for the family is two parents, a father and mother, raising children together, and that design must remain the goal for every Christian. To intentionally seek to establish a single-parent family is a departure from God's design and is therefore sin. Furthermore, single parenting is not wise, given its inherent challenges, and it is rarely if ever in a child's best interests when a two-parent family is a viable alternative for that child. Therefore, a single man or woman ought not pursue the adoption of a child prior to marriage under normal circumstances.
On the other hand, we can imagine circumstances when a single-parent adoption of a child might be a loving course of action, particularly in cases where the adoption can save a child from imminent abuse (assuming a two-parent adoption is not a feasible alternative).
Again, Scripture does not explicitly condemn this practice, so a Christian has liberty to act according to his or her own conscience, provided he or she is sincerely seeking to obey Scripture and is not acting contrary to any specific prohibition.