Control of departure from the Russian Federation is carried out at all border posts (checkpoints), regardless of the country of departure. And it does not matter whether you are flying by plane or traveling by car, the presence of debts (or rather a ban on leaving established by bailiffs) is checked at the border both at airports and at ground crossings.
The border with Abkhazia is no exception, where the presence of debts during passport control is also checked.. Check all passengers of cars, buses, trains, as well as drivers. And it does not matter whether you are going on a one-day excursion from Sochi, to a tour on a ticket, or it is an independent trip by car. If you have a ban on leaving Russia, established by the Federal Bailiff Service (FSSP), you will simply be deployed and sent back. You can't pay off debts on the spot and remove the restriction (see is it possible to pay off debts at the border).
But it is important to understand that for non-entry to Abkhazia, it is not the very existence of debts that is important, but the restriction on leaving abroad established for debts, i.e. a ban. Such a ban is established by the Federal Bailiff Service (FSSP), and only if you have accumulated a certain debt limit of at least 30,000 rubles for any debts, or 10,000 for alimony debts (see for more details, the amount of debt for a ban on traveling abroad). Therefore, before traveling to Abkhazia, if you have doubts that you may not be released, you should take care of paying off debts and removing restrictions.
If you managed by some miracle to pass the Russian checkpoint in the presence of a ban due to the inattention of the border service, you can enter Abkhazia on the Abkhaz side of the checkpoint unhindered, because there, of course, your Russian debts and prohibitions do not interest anyone.
In order to avoid unpleasant surprises during rest and travel, you can check yourself for debts before traveling to Abkhazia on the website gosuslugi.ru.