When is one absolved from paying in event of another punishment? Hope this setup makes it easy. This is a summary of אלו נערות Kesubos chapter 3 for those doing Daf Hayomi.
These two punishments have to be incurred simultaneously e.g launching an arrow 4 amos (cubits) on Shabbos in the reshus harabim (public domain) and tearing clothes on its path (Kesubos 30b-31a).
There are 4 types of punishment enacted by Beis din
3. Mamon - payment of damages
4. Knas - payment of a fine
Death together with payment of damages,* one is exempt from paying the damage when liable to death penalty according to everyone as the questioner brought from Shemot 21:22-23.
Death together with lashes one is exempt from lashes when liable to death according to everyone. See Kesubos 37a-37b where Rabbi Meir and Rabbanan both agree.
Death with Knas (payment of fine) Rabah in the name of Rabbi Meir holds you are chayav both, but Reish Lakish in the name of Rabbi Meir holds you are absolved from Knas when Chayav death. Rabanan hold like Reish Lakish whom the psak follows. (Kesubos 34b)
Lashes together with payment of damages ** Rabbanan (whom the psak follows) and Ulla in the name of Rabbi Meir both hold one is absolved from payment of damages when liable to lashes (except chovel and eidim zommemin for Rabbi Yochanan, and also seducing ones sister for Ulla,where one pays damages and is absolved from lashes), Because of "Kedei Rishato"- Only one punishment can be enacted (Devarim 25,2). But Rabba in the name of Rabbi Meir says one is liable to both lashes and payment of damages (Kesubos 34a-34b see Tosfos "Sovar").
Lashes together with Knas (fine) Rabbanan say one is absolved from the fine when liable to lashes with the exception of Motzi Shem ra where one is liable to both lashes and a fine of 100 shekel (with Motzi shem ra Rabbi Yehuda says you are only liable for the fine but absolved from lashes) (see Kesubos 46a) however Rabbi Meir says you are liable to both lashes and payment of fine e.g eidim zomemin (Makkos 4a).
Mamon (payment of damages) and Knas (payment of a fine) are both liable simultaneously according to everyone see Kesubos 29b e.g gneiva, or naara shenispatsa.
*Note: We pasken if there is no death penalty - because there is no warning or no beis din with semicha like nowadays - one is still absolved from paying (Kesubos 35a tana devei Chizkiya).
**Note:We pasken like Rabbi Yochanan who says that if there is no lashes - in a case where there is no warning or no beis din with semicha like nowadays - one is chayav to pay for damages, but Reish Lakish says even without enforcing lashes one is absolved from payment (kesubos 35a-35b)
P.S since we hold that chayvei Kareisos get Malkus when warned in front of eidim (Makos 13a), Rabbi Nechunia Ben Hakana (Kesubos 29a-29b) is effectively saying the same as Rabanan that one is absolved from payment in such a case of kareis (for which one gets Malkus so does not pay damages). However where there is no warning so Kareis happens without lashes, Rabbi Nechunia Ben Hakana holds one is exempt from payment but Rabanan would hold one is chayav payment like Rabbi Yochanan*