Here is an explanation of the secret engagements in S&S and Emma.

Marriage at a time when divorce was almost impossible was a very serious business.

For women it was the only respectable ‘career”, for as Charlotte Lucas muses in P+P:
…marriage had always been her object; it was the only honourable provision for well-educated young women of small fortune, and however uncertain of giving happiness, must be their pleasantest preservative from want.
Engagements were also very serious things to enter into, not to be done lightly, for there were serious consequences to face, including possible court action, if one party wished to break that contract without the agreement of the other party, and be set free to marry someone else.

So entering into a secret engagement was not a prank, or particularly romantic thing to do, as it may be seen today: it could have had serious legal consequences, as Edward Ferrars discovered when he became secretly engaged to Lucy Steel at a very tender age.

He could not back out of that “contract to marry” with Lucy without her consent, and so was stuck with her (until she decided to latch onto his brother, thus freeing him to marry Elinor).

For Frank Churchill persuading Jane Fairfax to accept a secret engagement was also a risky business. Displeasing his “adoptive” mother could have meant disinheritance (for his mother, Miss Churchill ,was lucky in that she was in possession of her own fortune and when he family cast her off she still retained some financial independence). If Frank were disinherited as a result of the wrath of Mrs. Churchill he would have had to rely upon his father for his financial future (and that would most certainly be not as spectacular as the riches the Churchill’s could bestow on him. I strongly suspect him to be like his mother:

She had resolution enough to pursue her own will in spite of her brother, but not enough to refrain from unreasonable regrets at that brother’s unreasonable anger, nor from missing the luxuries of her former home. They lived beyond their income, but still it was nothing in comparison of Enscombe; she did not cease to love her husband, but she wanted at once to be the wife of Captain Weston, and Miss Churchill of Enscombe.
Chapter 2, Emma

I think Frank very much wanted to be the heir to that great estate in Yorkshire AND he also wanted Jane Fairfax. He wanted his cake and to be able to eat it. Of course when he became secretly engaged to Jane, he had no idea that as a child of fortune, his path to matrimony would be made clear with the timely death of Mrs. Churchill, which removed all possible objections. He knew consent for his engagement would not be immediately forthcoming while his aunt lived, and so he was very wrong to enter into an engagement with Miss Fairfax, being unable to secure her an honest and financially secure future.

In addition to all these legal and financial facts to consider, there was also the problem of being deceitful and having to lie to all and sundry. In both Edward Ferrars and Frank Churchill’s cases, this leads to heartbreak for many of the parties concerned. As Emma comments in Chapter 46:

“Well,” said Emma, “I suppose we shall gradually grow reconciled to the idea, and I wish them very happy. But I shall always think it a very abominable sort of proceeding. What has it been but a system of hypocrisy and deceit, espionage, and treachery? To come among us with professions of openness and simplicity; and such a league in secret to judge us all! Here have we been, the whole winter and spring, completely duped, fancying ourselves all on an equal footing of truth and honour, with two people in the midst of us who may have been carrying round, comparing and sitting in judgment on sentiments and words that were never meant for both to hear. They must take the consequence, if they have heard each other spoken of in a way not perfectly agreeable!”

Frank Churchill compounds his culpability, IMVHO, by showing attentions to Emma, while secretly engaged to Jane Fairfax: he is lucky in that she does not fall in love with him, but he had no certainly that that would not be the case when he used Emma as a foil to distract Highbury’s attentions from him and Jane.

Jane Fairfax, an honest girl was also shamed by their joint conduct:

“On the misery of what she had suffered, during the concealment of so many months,” continued Mrs. Weston, “she was energetic. This was one of her expressions. ‘I will not say, that since I entered into the engagement I have not had some happy moments; but I can say, that I have never known the blessing of one tranquil hour:’ — and the quivering lip, Emma, which uttered it, was an attestation that I felt at my heart.”

“Poor girl!” said Emma. “She thinks herself wrong, then, for having consented to a private engagement?”
“Wrong! No one, I believe, can blame her more than she is disposed to blame herself. ‘The consequence,’ said she, ‘has been a state of perpetual suffering to me; and so it ought. But after all the punishment that misconduct can bring, it is still not less misconduct. Pain is no expiation. I never can be blameless. I have been acting contrary to all my sense of right; and the fortunate turn that everything has taken, and the kindness I am now receiving, is what my conscience tells me ought not to be. Do not imagine, madam,’ she continued, ‘that I was taught wrong. Do not let any reflection fall on the principles or the care of the friends who brought me up. The error has been all my own; and I do assure you that, with all the excuse that present circumstances may appear to give, I shall yet dread making the story known to Colonel Campbell.”
Chapter 48