Influenza A virus (IAV) is a single-stranded, negative-sense RNA virus and a common cause of seasonal flu in humans. Its genome comprises eight RNA segments that facilitate reassortment, resulting in a great variety of IAV strains. To study these processes, the genetic code of each segment should be unraveled. Fortunately, new third-generation sequencing approaches allow for cost-efficient sequencing of IAV segments. Sequencing success depends on various factors, including proper sample storage and processing. Hence, this work focused on the effect of storage of oral fluids and swIAV sequencing. Oral fluids (n = 13) from 2017 were stored at −22 °C and later transferred to −80 °C. Other samples (n = 21) were immediately stored at −80 °C. A reverse transcription quantitative PCR (RT-qPCR) pre- and post-storage was conducted to assess IAV viral loads. Next, samples were subjected to two IAV long-read nanopore sequencing methods to evaluate success in this complex matrix. A significant storage-associated loss of swIAV loads was observed. Still, a total of 17 complete and 6 near-complete Polish swIAV genomes were obtained. Genotype T, (H1avN2, seven herds), P (H1N1pdm09, two herds), U (H1avN1, three herds), and A (H1avN1, 1 herd) were circulated on Polish farms. In conclusion, oral fluids can be used for long-read swIAV sequencing when considering appropriate storage and segment amplification protocols, which allows us to monitor swIAV in an animal-friendly and cost-efficient manner.