What's the scriptural basis for Protestants to believe that their future sins have been forgiven?

Clarification

You described being "fully forgiven" as when past, present and future sins are forgiven. While I understand what you mean by this term "fully forgiven," some Christians who do not believe future sins are forgiven might actually describe the forgiveness they received as being "full" as well. It might be best to come up with a better term than "fully forgiven" since it seems that almost all Christians could describe themselves by this term in some sense.

The Answer

The strongest argument from scripture which Protestants use to prove that their future sins have been forgiven comes from Colossians 2.

And you, who were dead in your trespasses and the uncircumcision of your flesh, God made alive together with him, having forgiven us all our trespasses, by canceling the record of debt that stood against us with its legal demands. This he set aside, nailing it to the cross. (Col 2:13-14)

From this point, some Protestants insist that "all" really does mean all, in the sense of all past, present and future sins.

Conclusion

There are other proof texts some Protestants will use, but these are usually dependent on their specific theology. For example, some will use some scriptures to argue for a Calvinistic soteriology from which it logically follows that all past, present and future sins are forgiven. I chose Colossians because it is probably the strongest verse to argue for this concept without having other theological dependencies.

Disagreeing denominations have other ways of explaining Paul's meaning here, but I will not explain those interpretations as that is outside the scope of the question.

I cannot speak for all Protestants, but I did find this quote from John Stott, which explains the past, present and future aspects of the forgiveness of sins for those who have come to saving faith in Christ Jesus:

I have been saved – in the past – from the penalty of sin – by a crucified Saviour: “For in this hope we were saved" (Romans 8:24). "For I delivered to you as of first importance what I also received: that Christ died for our sins in accordance with the Scriptures" (1 Corinthians 15:3).

I am being saved – in the present – from the power of sin – by a living Saviour: "For the message of the cross is foolishness to those who are perishing, but to us who are being saved it is the power of God" (1 Corinthians 1:18).

I shall be saved – in the future – from the presence of sin – by a coming Saviour: "God shows his love for us in that while we were still sinners, Christ died for us. Since we have now been justified by his blood, how much more shall we be saved from God's wrath through him!" (Romans 5:8-9) "For our sake he made him to be sin who knew no sin, so that in him we might become the righteousness of God" (2 Corinthians 5:21).

That is not to suggest that a Christian can continue to sin with impunity. First John 3:6-9 clearly states that a true Christian will not live in continual sin. In response to the accusation that grace promotes sin, the apostle Paul declared:

"What shall we say, then? Shall we go on sinning so that grace may increase? By no means! We died to sin; how can we live in it any longer?" (Romans 6:1-2).

"In the same way, count yourselves dead to sin but alive to God in Christ Jesus. Therefore do not let sin reign in your mortal body so that you obey its evil desires. Do not offer the parts of your body to sin, as instruments of wickedness, but rather offer yourselves to God, as those who have been brought from death to life; and offer the parts of your body to him as instruments of righteousness. For sin shall not be your master, because you are not under law, but under grace. What then? Are we to sin because we are not under law but under grace? By no means!" (Romans 6:11-15)

"No one who lives in him keeps on sinning. No one who continues to sin has either seen him or known him" (1 John 3:6).

When Christians sin (as we all do), we are to confess our sins before God:

"If we confess our sins, He is faithful and just and will forgive us our sins and purify us from all unrighteousness” (1 John 1:9).

The concept of relational forgiveness is based on the fact that when we sin, we offend God and grieve the Holy Spirit:

"And do not grieve the Holy Spirit of God, by whom you were sealed for the day of redemption" (Ephesians 4:30).

That is why we confess our sins to God, not to maintain our salvation, but to bring ourselves back into close fellowship with the God who loves us and has already forgiven us. I hope this goes some way to explaining the Protestant view of sins past, present, and future, being forgiven on the basis of the shed blood of Christ Jesus and His righteousness being imputed to us.