This is an area where I struggle from time to time, so I'd be interested in hearing from the BB community regarding the ways they would counsel/advise someone experiencing this lack of confidence.
I think much of the anxiety comes from the common idea that salvation is a kind of transaction that occurs invisibly, and usually provides no evidence that it has occurred. The teaching of Jesus and the New Testament writers tell a different story.
On a theological level, the epistle of 1 John tackles this subject directly and lays out how one can know they are of the faith (1 John 5:13). It lays out the differences that characterize one who is simply religious and one who is of the faith in stark terms. When reading it, the reader needs to be careful to look at the differences in a broad sense, rather than the specific sense. Also, the modern Western Christian needs to keep in mind that the modern Baptist and evangelical churches - for the most part - have not taught practical sanctification and how to overcome sin. So, for instance, some readers read 1 John 3:19 and see that "no one who is born of God will continue to sin" and assume that they have not been saved, when instead, they simply haven't been properly discipled. If one is grieved over persistent sin, that is evidence that there is something real going on. But instead of being grieved, you need to take action.
In practical terms, the way you can gain confidence of salvation is to simply become a disciple of Jesus and, as part of that process, undertake practical discipleship and learn how to overcome sin. To have any success, you must have the aid of the Spirit. If you ask for divine assistance in following Christ, you will receive it. Through the process of discipleship you will find confirmation that you are living in the kingdom of God.