Lately, I have noticed brown stains on my laundry. I do about 2 loads a day and my washer and dryer are about 10 years old.I am guessing rust stains-Any suggestions, besides buying new appliances?

1) Run the cold water Faucet/tap into a white pan ect. through a piece of kitchen towel, and see if there is any discoloration? the towel will pick up any "grit" in the water supply (and will turn brown if there is any "earth" contamination)
2) Run the same test on the "Hot" supply
3) Get somebody to check the filters on the inlet hoses to the machine in ten year they have either punctured allowing all sorts of muck through into the machine (they are cheap to buy from a hardware store),
this is probably whats happened to your machine...
There may be a sludge trap fitted on the machine,if this has one it too could be blocked/restricted syphoning muck from your sink back into the system ,(not unheard of ) causing a "stained" wash (replace this filter & flush system). This could be a plumber job? but If it was me I'd unhook the machine take it outside and thoroughly flush and backflush the system through with a garden hose taking all the filters out beforehand then I'd put all the filters (new ones if possible)back in, reconnect up all the pipe work , then get a bottle of "descaler"(the stuff for cleaning your coffee pot or whatever? pour it into the "Drum" (not the detergent drawer) set the controls for a Hot wash,and set the machine going with nothing in it then as soon as it gets to the first "wash cycle" stop it then let it stand for a few hours
Then, set the machine controls to where it "drains" the water and thats it, your drum should be like new the heater element should be clear of any limescale,If you live in a "hard" water area contamination of heater elements happen a lot, and this is the best method of dealing with the problem.Ask the water supply people this question(and they should have a leaflet?)
Report back here if what I have suggested makes an improvement?