If they are fighting each other when they get excited over something, it sounds like redirected aggression. Spaying will help with this, but IMO, several things could going on.
As you're willing to look into training, that is good. SOME of it could be a dominance issue.
I wonder though, if the incidences might go something like this.
Y1: Oh look a SQUIRREL!! OMGOMGOMG GET IT!!
Y2: OMGOMGOMG!!!
Y1: I can't get it!!! *frustrated*
Y2: Me either *frustrated*
<yorkies fight>

or even possibly
Y1: LOOK A SQUIRREL!!! Get it!!!
Y2: Oh, what? yeah yeah!!! OK calm down
Y1: No I want to get it!!!
Y2: I SAID calm down!!
Y1: you're not the boss of me!!
Y2: Oh really?
<yorkies fight>

The first, is obviously the redirected aggression, the second is about dominance. Spaying can calm them both down, but dominance and aggression is not always about hormones.
You will have to determine which is happening with your dogs that causes the fights, or if it is a combination of both.
I am a Cesar Millan fan, and I do believe that all dogs should be calm submissive to their owners and to their pack. Because of your pack dynamic, I am almost willing to bet what is going on is more the second situation than the first, with Lucy trying to discipline the younger Ellie and Ellie having none of it.
And yorkies being TERRIERS, they can have a zest at all things distructive, including fighting each others (nothing worse than breaking up a terrier fight, I remember having to separate a pack of Jack Russells that got themselves into a NASTY free for all).
Good luck with your girls.

I doubt that you will have to rehome anyone, this can be worked out in almost every situation. When you sperate them do NOT give them the pleasure of people company. You are the pack leader and do not have to allow fighting. If they fight put one in a room with a gate or door closed. The other same thing time out away from people and pack. I think 5 minuets is about right. I warn mine with a firm ENOUGH and they knew after only a few days that I meant business. Also when breaking up a fight, it good to toss a pillow, rolled up socks etc. at them first to break their concentration. Because when they are in full fight mode they truly do not hear you. This also helps prevent you from getting accidental bites. Like spoiled children Yorkies will try to divide the pack to get what they want. Don't let the little stinkers get away with it. YOU are ALWAYS the ALPHA. Hope this helps.
I do agree with reading Cesar Millan books and videos. His approach often works out many problems, because he really understands how dogs think.