Are there any services translated to the Lemko/Rusyn/whatever name you call your language? Or "religious literature"? Ruthenian Catholics also are kindly asked to answer.
"Po nasemu" is a politically safe to use for the name of the language when speaking to any American born or raised Rusyn, Lemko etc... But since in the USA, 'po nasemu' has pretty much become American English since the late 1960's in both the Orthodox and BCC churches of these people, there is not, to my knowledge, anything close other than the 'proper' Ukrainian language texts used by UOC-USA and the ones used by the UGCC USA. Neither the BCC nor ACROD parishes use the Ukrainian in the US. In Europe the Orthodox use the Slavonic while the Greek Catholics in Slovakia use a combination of Slovak and Slavonic while those in Transcarpathia predominately use Slavonic with some use of vernacular Ukrainian. Some folk hymns were always in the vernacular, but again, English has supplanted those as well in the USA.
As to literature, the first half of the twentieth century did produce a great volume of religious treatises and polemical material in the Rusyn/Lemko literary style popularized by Father Alexander Duchnovych in the 19th century - again tilting to a combination of then-contemporary 'po-nasemu' together with some formal Ukrainian, Russian, Slovak and Magyar idioms. The internecine religious disputes within the American Greek Catholic communities which lead to the creation of what is now OCA, UOC-USA and ACROD produced much in terms of published material most of which is regarded today as 'over the top' propaganda and polemic.
A good starting source would be the library collections of Christ the Savior Orthodox Seminary (ACROD) 225 Chandler Ave, Johnstown, PA USA 15906 and of SS. Cyril and Methodius Byzantine Catholic Seminary located in Pittsburgh, PA. Also, the Carpatho-Rusyn Society archives, located in Pittsburgh, PA host a large collection of such works - from both the Greek Catholic and Orthodox factions. Some of the rhetoric is, for lack of a better word, outrageous and entertaining to modern ears. If you want I will PM you how to contact the CR-S President.
Hope this helps.