Hiya, sorry to hear about your problem.
There will be a skin realted problem that your beagle has. Has any diagnosis been attempted?
Sometimes cats and dogs can have allergies to anything really - grass , nettles, deodrant etc. It may have been something like these things i am mentioning . Try and think back have you used any shake and vac for instance? or a new washing powder .....
It is best to keep a diary that you can refer to everytime this happens and use a process of ellimination to try and prevent this happening again.
Please think at all angles such as food, shampoo etc.
It can be normal for dogs noses to become dry from time to time , i routinley advise owners to put vasaline gel on there to keep it moist. Especially in older animals to prevent it cracking and becoming dry.
If you wanted to find out the problem then a skin scraping or a small biopsy can be sent away to the laboratory for analysis, if the results come back normal then that would be nice to know aswell so money not wasted.
Skin testing can be perfomed on dogs and cats by a referal dermatologist that are better experienced in one feild or expertise , please ask your veterinary surgeon if you are considering taking this path.
Good luck and let me know how you go on. xx