I would definitely get blood work done at the vet. Its odd for a 4 year old bird to have an overgrowing beak unless there are underlying liver problems. And seed only diets tend to cause liver problems. Most often on a young bird with no other issues, this can be resolved with a diet change.
The cage is probably okay unless its really rusty or has dangerous bar spacing where she can stick her head through or places wher she can get toes stuck. Also check it over for sharp ends of wires as some cheap small mammal cages are like that.
You really need to go out this weekend and get a digital gram scale. You can buy one for about $30-$40 at an Office Depot or similar. Whenever attempting a diet change, you need to watch the birds weight. Do not attempt a diet change unless you have already been weighing the bird for a few days and you have established what normal is for your bird. You also need to find out from your vet if your bird needs to loose any weight. Often birds with overgrown beaks are also a little overweight which leads to fatty liver disease(hepatic lipidosis). Although from the pics your bird doens't look too bad.
When I start a diet change on a new bird, I start by adding new things but not taking away any old things. That is something you can do now. Pick a pellet(my birds like Zupreem Natural pellets in the parrot/conure size). Put a few pellets in her bowl with her seed. A lot of birds that have been on seed only for a long time will convert to pellets faster than veggies because they are dry and some birds don't like the icky slimey feeling of wet food if they aren't used to it. Getting your bird on a mostly pellet diet is probably the quickest way to solve the dietary issues leading to the overgrown beak unless you can get her eating veggies faster.
After a couple weeks, once I'm sure the bird's weight has stabilized and I have a good record of what normal is and I know that the bird is at least at a healthy weight, I will start reducing the seed and increasing the pellets. If the bird is a hard core seed junky, they will often resist this by loosing 10-20 grams over a couple of days, acting unhappy, pacing and generally being pathetic trying to make you break down and give the old food. Amazons usually aren't that bad to convert. Once you get past those 2-3 rough days where they think you are for sure starving them to death, they will start eating their pellets with gusto and not look back unless you start giving them so much seed they feel full enough to be picky again. You do the same thing with veggies. You can offer now, but later on while your converting to pellets, have a bowl of veggies in there too so the bird has a choice of veggies and pellets. Nether is a bad choice. Down the road, it would be good if at least 25-50% of the birds diet is fresh or fresh and cooked at home veggies. IMO pellets and seed only can lead to kidney problems later on. They need the moisture that real food brings. But for short term, pellets might be a quicker fix to the dietary problems causing your bird's beak issues.
Also for reference since a lot of new people are convinced their bird isn't eating hardly anything, my 500 gram yellow naped amazons get about 1-2 tablespoons of pellets. 1 teaspoon of seed. And another 1-2 tablespoons of a combination veggie, fruit and cooked or sprouted bean and grain mix. So about 3-5 tablespoons total of food per day depending on how dense the soft food is. Most likely your red lored is smaller than my birds, so will eat less. In other words, if its cage came with a large bowl and you fill it half full and wonder why it doesn't look like the bird is eating much, probably you gave it way more food than it would normally actually eat. I just mention that because some people become alarmed that the little angel surely can't be eating nearly enough to live when doing a diet conversion.
Also, amazons do have a certain aroma that is natural for them. If they are stressed(scary new home) or sick(bad diet) they will tend more toward a dirty socks smell. If healthy and happy, it tends more toward a nice homey sweet musky smell. Either will take getting used to, but I think once she calms down, gets a few baths and you get her diet straitened out, it won't be so noticeable. It may take a couple months though to get her skin feeling better if she had a vitamin A deficiency which almost all seed only amazons do. You'll probably notice the smell is super strong at the vet and in your car on the way home.
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At least not to this nose.
