This wasn't some secret that families tried to hide because of the shame. Everyone knew what was going on in every level of society, and no one cared. If anyone in the country was responsible for half the bastards running around, it was the royal family. At one point, only one of Mad King George's 13 children was legally married, yet he had at least 19 and possibly as many as 56 illegitimate grandchildren. And it wasn't like these kids were hidden away; his sons openly lived with (some of) the mothers of their illegitimate children, and it was common knowledge and reported in the press.
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If you didn't want that level of commitment, there were always the brothels. Prostitution was not only completely legal, guides were published every year informing men of where to find them, and what sort of things they were willing to do, like a Michelin Guide for ass. By one historian's estimate, in London alone prostitution was the equivalent of a $2 billion a year business.
If it turned out you couldn't stand your partner after you married them, no problem; affairs were unbelievably common as well, especially among the upper classes. Spouses just turned a blind eye to the other's philandering. The Duke of Devonshire even moved his mistress into his home and lived openly with both her and his wife for 25 years, and everyone in the country knew about it. But women gave as good as they got, and the Duchess gave birth to a daughter she conceived during one of her affairs with the future Prime Minister Charles Grey. That would be like if Angelina Jolie cheated on Brad Pitt and had Barack Obama's baby AND NO ONE GAVE A SHIT.