If you want to dry-hop, you have to deal with solids. Hops are solid.
Worrying about sterility is needless worry. As RSmith rightly points out, not only is beer a fairly inhospitable place for any contaminant you're likely to inoculate with hops, brewers have been dry-hopping with loose cones for 150 years and with pellets for as long as there have been pellets. If you're that
worried, steep the pellets in 180+F water for at least 10 minutes, maintaining the heat at or above 180F.
Your vodka technique won't really extract the oils and fatty acids you impart by letting bits of hops flowers soak in beer. You'll be adding more alcohol, but precious little in terms of flavor and aroma. Hops extraction is a much more complicated process. And even if you could
perform carbon-dioxide extraction, blind-taste panel data suggest that the flavors and aromas imparted by hops extract are inferior to those imparted by old-fashioned chucking hops into beer.
Just add the dry-hops and don't worry about it. Let it be part of the magic! Science is important to brewing, but part of it is magical; just ride on that. You'll be happier.