If you DON'T have the "advanced" licence, there is a workaround but it will not be exactly the same result.
It starts the same way : From a cut you could have multipart polygons, which are more difficult to manage. Therefore you should first use "multipart to single part", then select the small polygons (select by location all polygons touching the one you used for cutting, then use a size threshold).
Then export the selected polygons in a new layer, inverse the selection and export the other polygons in another layer. Now you have 2 distinct sets : one with the slivers and one without the slivers.
Use spatial join to get the OID value of the non sliver polygon that is closest to each sliver polygon.
Go back to your initial layer, add a new "temporaryOID" field and join by attribute the slivers'table to the initials'table. Update the temporaryOID based on the joined "not sliver" OID of the sliver polygons and the "not sliver OID" from the non sliver polygons.
Run the dissolve tool.