What you are describing is very common. The root cause of this issue is the occlusion of the cartilaginous outer part of the ear canal. When the ear canal is occluded lot of the low frequency energy components of your own voice, and other sounds produced in the mouth (like chewing), are not able to escape like they normally would through the opening to the ear canal, and a greater amount of low-frequency acoustic energy will reach the eardrum.
To my knowledge, Signia is the only manufacturer which has actually implemented a technological solution which addresses this problem without causing negative knock-on effects. The technology is called Own Voice Processing. During fitting, the hearing aids are trained to recognize the wearer's own voice, and then it is de-emphasized during routine use. At this time it is only available in their Receiver-in-the-canal hearing aids, such as the Pure 312 Nx. So if you need to wear a custom ITE hearing aid, this would not help you.
To be clear, open fittings and other occlusion reduction strategies will have knock-on effects which will usually result in poorer speech intelligibility.
I would first contact your hearing specialist and setup an appointment. Explain to them the issues you are having. They will be glad to set down with you and make the needed adjustments either in the programming or possibly the ear piece. There are may things that can be done to eliminate or listen the plugged feeling. It may take experimenting in the office but can be done easily.
Yes, hearing one's own voice in an unusual way is not uncommon. Your audiologist is able to program some of those "undesirable" sounds out, but continual wear of the hearing aids will allow you to adjust to hearing your own voice again. The sense of nasality and echoing often signals that the hearing aids are cutting out too many outside sound that usually are naturally going into your ear canal. If you pull the hearing aid out slightly, does it still sound echo-y? If it is less, then the audiologist can adjust the shape of the shell to better accommodate you.
An "echo" in your own voice is usually due to one of two things - occlusion or ampclusion. Occlusion, as others have mentioned, is the result of the hearing aid "occluding" too much of your ear canal relative to your hearing loss. This can be improved/resolved by opening up the venting of the part of the hearing aid that is in your ear (i.e. dome, earmold, or the physical shell of an in the ear hearing aid). Ampclusion occurs when we amplify more sound than what someone is accustomed to, and typically will go away with time - but the best way to resolve this is by starting the hearing aid user off at a lower setting and then slowly increasing the volume of the hearing aids until we get to their hearing loss prescription. In this case, the auditory system needs time to acclimatize to more stimulation.
The individual who fit your hearing aids should be able to determine if this is occlusion or ampclusion fairly easily if they fit the hearing aids properly - either way it is something that can be improved. Real ear measurements can also be very helpful for measuring occlusion (as well being an essential part of the hearing aid fitting in general).
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