I don't know if there is a relationship between Sjogren's and sleep apnea, but there are ways of coping with dry eyes and dry mouth while using CPAP or related treatment equipment for sleep apnea. There are many types of masks, so if you had difficulty with the one you tried at the sleep study, there are likely to be others that will be more comfortable for you.
I also have a problem with dry eyes and dry mouth. I use preservative-free eye drops before going to sleep and sometimes during the day. There are a number of brands over-the-counter, as well as prescription Restasis, in case you have not tried any of these. Treatment for sleep apnea should not make dry eyes worse but you should mention your symptoms to anyone prescribing sleep apnea treatment for you. As for dry mouth, if you already have dry mouth before starting CPAP, your physician should prescribe a humidifier be included with the CPAP machine. That will help somewhat. If you use a nasal pillows or nasal mask that does not cover the mouth, you need to keep your mouth shut so your throat does not become dryer due to the mask I use a nasal pillows mask and find that the air from the machine helps keep my mouth shut, but some people need a chin strap with such masks to keep their mouth closed. A room humidifier and drinking sufficient water also help me. Another option is a full face mask that covers the nose and mouth, which is essential if you breathe through your mouth. You should try on at least one of each type of mask--full face, nasal, and nasal pillows--while lying down so you can find out which type is most comfortable for you. Within each type of mask, there are a number of models. A durable medical equipment (DME) can help you determine which model of mask is most comfortable for you. Every face is different and different people prefer different masks.
If you are not being treated for Sjogren's, I would get a diagnosis and treatment for that, as well as the cause of the neuropathy. I believe Sjogren's is treated by rheumatologists. I was diagnosed with it once but not in subsequent tests.
If you have not gotten a detailed copy of your sleep study results, be sue to ask for a copy to provide to your physicians and also to keep for your own use as you learn more about sleep apnea. Did the sleep study indicate restless leg syndrome as well as sleep apnea?
There are a variety of types of CPAP machines and masks, so if you didn't like the one used during the sleep study, there are other types of both. A key to successful treatment is finding a combination that fits your particular needs and that is comfortable for you. It is a hurdle we all have to go through and sometimes finding the right combination involves trying a number of different masks.
Since you said your sleep apnea is mild, another treatment option that may help you would be an oral device that keeps your airway open. It should not affect your eyes. If you decide to explore that route, I would recommend finding a dentist that specializes in treating sleep apnea. I have used an oral device for TMJ problems. A custom-made device fits over the teeth and can be quite comfortable during sleep. Usually, oral devices alone are not considered sufficient for severe sleep apnea but they sometimes are effective in mild cases of sleep apnea. If an oral device sufficiently treats your sleep apnea, you would not need to use CPAP-type equipment with a mask at all.
The most important thing is for you to find a treatment that you can use every night so your conditions do not get worse. The right treatment will make your sleep much more comfortable and improve your overall health. Don't despair. There are many options for treating sleep apnea and no single solution fits everyone. You were given only one option to try during the sleep study, it appears, and there are many others.