Does asking for forgiveness exonerate you from bad karma?

Certainly yes!
but provided it is done with true repentance in our heart and a firm resolve to not do that again.

If one wishes to be absolved from bad karma, then ‘Pratikraman’ is the assured path for that
Pratikraman means:
To confess before God that ‘I have made this mistake’,
With genuine repentance in our heart, seek forgiveness from God, and
Ask for strength from God for not doing the mistake ever again.
This is called pratikraman!

Whenever we hurt any living being through the mind, speech or body, bad karma gets created. For every such instance, we must do pratikraman with true repentance in our heart. When tea spills over our dress, how quick we are in removing the stain? Similarly, the internal stains of karmas can be removed by doing pratikraman. The sooner we do it, the better it is. Hence, as soon as we realize, ‘Oh! I made this mistake’, we must do pratikraman.

Param Pujya Dada Bhagwan, the Enlightened being, explains,
“If bad karma ends up being done, then there would certainly be a solution for it. Always, when bad deeds happen, repentance must immediately follow, and you should repent wholeheartedly and with sincerity. Despite repenting, if the mistake is repeated, then do not worry about it. You should repent once again. You are not aware of the science behind this; that is why you may feel that despite repenting, it is not coming to a stop. Why it is not coming to a stop is a science. Therefore, you should simply carry on repenting. For the one who repents wholeheartedly, all his karma are washed away. If someone feels hurt by you, then you should definitely repent over that.

If atonement does not happen, then more karma gets bound. By atoning, the karmic tubers become lighter. Otherwise, the result of that demerit karma is terrible. One may even forfeit human life [for the next life], and if one ends up as a human [in the next life], then he would face all kinds of difficulties, that of food and water; and he would certainly never get any respect; constant insult. That is why this atonement and other rituals need to be carried out. This is referred to as indirect worship. As long as one does not attain Self-realization, it is necessary to engage in indirect worship”

And in whose presence should one ask for forgiveness?
Whomever you have faith in. “Whether you believe in Lord Krishna or in Dada Bhagwan, whomever you have faith in, do it with Him as a witness”, suggests Param Pujya Dadashri.

Here, a doubt may arise in our mind...
that on one hand, one may continue to carry out bad deeds, and on the other hand, he continues to repent – is that not pretense?
Param Pujya Dadashri clarifies, giving the simile of an onion, “That is not what should be done. A person who carries out bad deeds and repents over them, he is not able to feign repentance at all. That repentance is certainly real, and as that repentance is real, one layer of the ‘onion’ [karma] peels away. However, the onion still appears whole. Then another layer peels away. Repentance never goes to waste.”

And what in case if one is not asking for forgiveness wholeheartedly?
Param Pujya Dadashri assures, “The one who asks for forgiveness certainly does so wholeheartedly. And even if you were to ask for forgiveness superficially, it would still be acceptable. However, you should definitely ask for forgiveness.”

He emphatically claims,
“If it becomes a habit, then let it be. But ask for forgiveness. Without asking for forgiveness, consider yourself to be in trouble!!! What is the meaning of [asking for] forgiveness? It is referred to as pratikraman. And what is a mistake referred to as? Atikraman.

If someone is drinking brandy and says, “I am asking for forgiveness.” Then I would tell him, “Ask for forgiveness. Continue asking for forgiveness and continue drinking. But decide in your mind that, ‘Now I want to give it up.’ Decide in your mind wholeheartedly, ‘I want to give it up.’ Then continue drinking and continue asking for forgiveness.” One day, it will come to an end. This Science (Vignan) of mine is of a hundred percent. This is, in fact, a Science!! It will not refrain from giving result. It gives instant results. This is the ‘cash bank’ of divine solution. This is indeed a ‘cash bank’! It has not emerged in the last one million years! It gives liberation within just two hours!! I am ready to give you whatever you ask for over here. One may feel tired of asking, but the Gnani will not be tired of giving.”

Pratikraman breaks the bondage caused by the ego, which is at the root! And when we come to Akram Gnani, he uproots this very root itself!
How? By destroying our ego and helping us attain Self-Realization with His divine grace.
He does this in just two hours time and puts us on the path of liberation.

Param Pujya Dadashri explains, “Carrying out good deeds is considered dharma (religion), and carrying out bad deeds is considered adharma. And to go beyond dharma and adharma is considered Atma-dharma (religion of the Self). When you carry out good karma, credit is created; you will have to come back to enjoy that credit [in the next life]. When you carry out bad karma, debit is created; you will have to come back to suffer that debit. And where there is no credit and debit in the ledger, the Self (the Pure Soul) is attained.”

Thus, asking for forgiveness indeed exonerates us from bad karma. And going to Gnani and attaining Self-Realization from Him, shall liberate us from all karma.

To know more: https://www.dadabhagwan.org/self-realization/