Ok let me get this out of the way first, yes I suck so flamers take ye touches else where I already know. I was just kind of wondering what the trick is,

I killed the high dragon at level 14 on my protagonist, 13 on my party on normal difficulty.  It was challenging and took a lot of potions and a spirit healer to get me through it.
 
My party was:
Protagonist:  Shield specced warrior.
Shale:  Specced for tanking.
Wynne:  Spirit healer, occasional dps.
Leliana:  Ranged dps, mana regen bard song.
 
Having upgraded shield wall is very helpful.  Shale and my warrior tanked the dragon, I started the fight tanking with my warrior and then moved shale around to the dragon's tale.  My healer and ranged dps stood out of the way so they would not get hurt when either tank was tanking.  Just took it slow and eventually the dragon fell and gave me 12 items of varrying usefulness.
 
For the other dragon, it is way easier, did it without even needing a single mana pot, not really sure how or why that happened but I do not question good fortune.

Morrigan with Cone of Cold saved me. 

Not going to read the spoilers but Roosterteeth suggests this: 

Cone of Cold is a BIG help. Freezes the dragon every time in my exp.

Cast the Aging spell, it says so right there.

Here is the way I did it. I picked one party member to tank (shale) and then my main character, leliana, and zevran all brought bows and plinked that dragon to death while shale chugged health and took all the blows. Cone of cold will do wonders, but this is the way I did it with no mage available which might be the case if you are fighting the shape-shifting Flemeth.

HEXES! They are your friend, wont work on a particular dragon where Morrigan is out of the question - but I suggest a regular tank & spank combo with a tank in Fire Resistance gear.
There's a hex to make all his hits becomes misses and crits regular hits, there's also a hex that'll make ALL hits crit on him. 

Using that will make tanking easy (Fire damage is your main concern from here on out) and damage will pick up as long as you use melee damage as your main damage output.

Combine that with the Primal Frost Chain that Morrigan is invested in makes her one of the best casters in the game for boss-slaying.

I ended up killing Wynne, so Morrigan is my only healer for both dragons.  At around level 12-13, I realized with Shale in tank mode and Large Fire Crystals on him as armor, he doesn't take very much damage from either dragon.  Used him to tank, Morrigan to heal with the rank 1 heal spell.  She wasn't 14 yet, so I wasn't a spirit healer with her.  The other two members of my party died early on in the fight.  Shale and Morrigan had little problems taking him out.
 
For the other dragon, that was a little weirder since I had no healer.  All my party died except for Shale when the dragon was around 70%.  Turns out I had about 30 lesser health pots, and that was plenty for Shale to solo her.  The fire resist is a must though.

Might be worth taking the drake scales to Denerim and getting the drake armour made before fighting the dragons? They have some pretty killer fire resists. I think the full set gives around 75%? I dunno, never tried it myself so it may be a complete waste of time.

Had 2 people with cone of cold on this fight, and had 3 people with heals. Kept one of my cone of colds in reserve so that when the dragon grabbed my tank in her jaws I could freeze her and have him fall to the ground without taking too much damage.
 
Course being 18/17 made this fight a bit of a breeze.

For the High Dragon, there is a narrow shaded area that the dragon can't get through.  She will fly from one side to the other, trying to get as close to you as she can.  Move so that you are constantly on the opposite side from her, and pump her full of arrows and magic.  Keep repeating until she is finished.
 
With Flemeth, I used a similar strategy, but sent one tank in to keep her in place.  I put on a greater warming salve for fire resistance, and then focused all my healing into that one person.  Then I had the other characters just attack with arrows from a distance.  Worked quite well.

THANKS FOR SPOILING THE GAME TELLING ME THERE ARE DRAGONS IN IT JEEZZ!
 
I kid, some good advice in here.

Seems like you need to be a reasonable level to beat this thing, I'm sure it can be done at lower levels but its a pain. 
 
My strat was Shield wall tank; 2-hand Indomitable warrior; Rogue Archer with Regen Bard song, Ranger Blight Wolf; Mage with Glyph of Warding cast on tank, Weakness on dragon, Heal and Group Heal, handful of Poultices for when stuff goes wrong.
 
I think i was about level 15