There are no restrictions on music being a cappella during the period from Rosh Hashananah (New Years) until Yom Kippur. It is simply a time of introspection and self evaluation, trying to find ways to make ourselves the best we can be and figure out how to change our ways from the things we did not do properly in the previous year. Some people might find that music makes them too lighthearted to be able to accomplish these tasks in the proper frame of mind, so they might prefer not to listen to music at all during this time period.
As far as instrumental/choral music, you might like what is called chazzanut. It is where a man with basically an operatic-quality voice sings different prayers. There are many such CDs available. Rabbi Helfgot is a very famous one in our times. You can just google Helfgot and see what comes up. I see several videos you can try out to see if this is your type of music. If yes, there are sites that sell primarily (if not exclusively) Jewish music, such as mostlymusic.com. I am sure you can have CDs shipped to you.
Two side comments about music – 1 – the 2 times a year when there are any restrictions on listening to live/recorded music are during the period from Pesach until Shavuot and then after that from the 17th of Tammuz until Tisha B’Av. All other times of year we do not have the same issues. 2 – The period immediately following Yom Kippur is known to be very joyous, Sukkot is 7 days of happiness – we say in the prayers “Zeman Simchateinu – the time of our joy”. These 7 days are followed by a final holiday of Shmini Atzeret/Simchat Torah, when we dance with the Torah scrolls in the synagogue – something very beautiful to see.
GL with finding your route back to your ancestral home.