I understand that you are concerned about Stash's nasal congestion, (particularly on one side), sneezing bloody mucous and nasal irritation and I'd like to help.
Sneezing, and nasal congestion are signs of nasal irritation.
With nasal irritation we can also see a nasal discharge which can be yellow, white, green, mucoid or even bloody in character. If the irritation is enough the mucosa becomes raw and we can see blood in the discharge. This is usually coming from both sides of the nose and isn't more then droplets. If this was a short term issue with your girl it may be related to a contagious upper respiratory infection. Some cats get sicker than others and young, healthy adult cats seem to tolerate them and fight them off better than very young kittens or older cats. An upper respiratory infection in cats is just like a cold in you and I, and these are commonly caused by a virus, the most common one being Herpes virus. These infections are easily caught by breathing in virus particles in the air from a sneeze or nasal or eye discharge from an infected cat. Some cats get chronic infections and in times of stress the infection comes out of hiding and causes symptoms.
But given that this has lingered for a month this is not likely to be a simple viral infection. I am concerned too as it seems to be worse on one side, and generally viral infections affect both sides equally.
In some cases with long term infections, especially if this is a fungal infection rather than a viral infection, or if it is a mass we can even see nasal bone destruction and swelling or changes in nose conformation.
With an uncomplicated upper respiratory infection I would expect that she would be much better in 7 to 10 days. Since it has been over a couple weeks and she isn't getting better and is now sneezing blood I think we need to look further or consider giving her an antibiotic such as doxycycline or azithromycin.
Diagnosis of a the problem behind these symptoms with a long term problem can be complicated.
We may need to perform radiographs of the nose and sinuses looking in the bones or full sinuses, nasal flushes and scoping to collect culture and biopsy specimens and sometimes blood titers to look infection (especially if we are suspicious of a fungal infection). I would also check her for viruses like feline leukemia and feline immunodeficiency virus. In an older cat I would also check for liver disease, hypertension and hyperthyroidism as those diseases can lead to nose bleeds.
Possible ruleouts behind these symptoms are bacterial, fungal, tick borne or viral infections, a foreign body in the nose, a polyp or mass in the nose or a tooth/teeth root infection or metabolic organ disease or hypertension. Rarely the cause is an inherited clotting disorder, but I would have expected her to have symptoms of that before now.
Cancer would be possible too.
Things that you can do that aren't too expensive include checking a complete blood count and biochemistry profile and T-4 to look organ disease and hyperthyroidism, and possibly a clotting profile. I would also recommend checking her blood pressure.
If those tests come back normal then looking for a tooth root infection, a foreign body or polyp with a nasal scope and radiographs of her nose and sinuses under sedation would be recommended.
If you decide that you cannot or would not run any further diagnostics, then trying an antibiotic or trying a course of steroids could be an option. Steroids will reduce inflammation and improve her comfort level. Though ideally steroids shouldn't be used long term, or without making sure that a fungal or bacterial infection isn't present, they may reduce inflammation enough to stop bleeding.
If she is still eating and drinking well and not lethargic too, then she may not need emergency care today. Keep her quiet, use drops of saline nasal drops to add moisture to her nose and help decrease irritation and have her rechecked as soon as you can.
But if you are noticing a large amount of blood, her gums and tongue look white or very pale pink in color rather than a nice bubblegum pink, and she seems very lethargic then an emergency visit is best.
Best of luck with your girl, please let me know if you have any further questions.