I am a hairstylist with over 10 years experience. Hair color removers are typically extremely easy, effective and very safe to use on your hair. I don't know what the other ladies are talking about, they must be thinking of bleaching removers. Even Sally's has one that works great. Removers are best used for fresh, deposit only color jobs that went horribly wrong. Since you just re-colored your hair, the color removal could be a safe bet, so long as it is one that isn't a bleaching method. If you are going to fix it at home, go to Sally's and get the removal.
All medium brown hair has red in it. The color has both blue, red and yellow pigments. So the reddish tint is part of your natural color. If the reddish is only after you color your hair, it is likely because you are choosing a color with too much red or gold in it. So choosing a color with a hint of ash is okay, I would more so recommend using a color with a N/neutral base since too much ash can make your hair turn greenish or even 'muddy' looking, especially since your ends are pulling dark. If you go to Sally's you can choose a color that has natural and ash, such that will look like a 6/NA or 6/01 or something along those lines. The 6 is the level brown I am assuming you are getting based o your description. But ask the ladies at Sally's for a color formula since they can see your hair. Use 10 or 20 volume developer depending on the color you go with, again, ask for help there.
Per the colors' timing instructions, paint the color on your scalp/roots and leave on for the duration of time, minus approx 15 minutes. Then, very slightly spritz your hair with water and comb it through. Take your excess color, and color the rest of your hair, to about the last inch or so of your ends. Leave on until the last 5 minutes, then comb that color through to your ends. It will help not get that muddy/too dark end color. The reason the ends are going too dark, is because the hair is older, has more damage and is more porous.
Or, you can go to a salon (a better choice really) and have it professionally color corrected, they can then give you tips on how to do maintenance touch-ups.