As you have rightly quoted, Jesus is very clear on the matter of divorce and adultery. He clearly stated that there is to be no divorce, and common-law relationship or marriage with a divorcee is the same as extramarital relationship.
The problem with divorce is that it does not break the bond of marriage, in the eyes of God. It only breaks the legal obligation and privilege of marriage. The two divorcees are still married, are still not free to remarry.
Having said that, the Church also realizes that many failed marriages have been invalid in the first place, or rather has never been intended to be a marriage as God revealed it. For instance, God intends marriage to be two parties freely and selflessly giving oneself to the other, that they are to be faithful to God and each other, and open to life. But if the husband never demonstrated he intended to be faithful, or the wife naively expects that she will be pampered and refused to have children, that marriage can be proven to be invalid, and so nullifiable.
It is for this reason that the Church provides the recourse to annulment, the canonical (legal) process by which a divorced marriage is proven to have never been validated in actuality. It is also because of this that annulment is not the Catholic version of divorce. Not all marriages can be annulled. Only those that lacks the essential character of marriage, and can be proven before the tribunal, can be annulled.
So for a divorcee, what to do if he or she falls in love again? There is no real problem to date, as long as he or she is remaining chaste. And if remarriage is a real possibility, the person should immediately consult the local diocese to assess the possibility of annulment. Meanwhile, don’t move in together and wait for the annulment before making any plans for marriage.