Sounds like a cognate of the Polish "osad" which comes from the verb "osadzać się" (a similar verb "osiadać" can also be used in some cases) with similar meanings of settling down, sedimienting, forming deposits etc. to the Slovenian verb (with an exception for sitting down of a person). The noun itself refers to various kinds of sediments and deposits in geology and engineering (including filtration - so substances similar to the coffee grounds), and to brown traces of coffee or tea on a dirty glass or a cup - but not to the thick layer of leftovers at the bottom.
(U)sesti (se) is the direct cognate to
(u)siąść (się), also
(u)sedati (se) and
(u)siadać (się). (I'm not sure if all the permutations are actually possible in Polish - in Slovenian we have
sesti,
usesti se,
sedati and
usedati se). All forms can refer to sitting down (as a person), but only the ones with u- work for sediments like in geology or indeed our coffee example.
Osadzać się reminds me more of
usajati se (the forms agree as far as the iotated
d is concerned) which means "to be grumpy and bark at people", at a glance it seems connected with
saje in the chimney, always in the plural (cf. Polish
sadza) which is a kind of sediment as well. I guess the motivation is that an annoyed person will spout out his complaints like soot is falling out of a chimney.