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The game doesn't consider such things when assigning damage. As always, you need to assign at least as much damage as would be required to be lethal damage (the damage receiver's toughness minus any damage already marked on it, or 1 if the damage dealer has deathtouch), then any excess can be assigned to the player.

Your friend is correct.

For damage assignment, you only need to assign lethal damage to all blockers before a trampler can assign damage to the defending player or planeswalker.

Lethal damage here means 1+ damage from a creature with deathtouch or enough damage (including present and previously assigned) to equal the toughness of the blocker from creatures without deathtouch.

Avacyn has an 8 toughness, therefore lethal damage (for damage assignment purposes) is at most 8 damage.
Your 10/10 can therefore assign 2 to her.

If Avacyn had already been damaged (say you Bolted her for three earlier this turn) only 5 damage would need to be assigned and you could trample over with 5.

Note: You choose the damage assignment of the 10/10 so you can choose not to have the 2 damage assigned to the defending player and assign it to Avacyn if you like.
eg. you attack Jace (3 loyalty) with a 5/5 trampling doublestriking lifelinker and your opponent blocks with a 1/1, you could trample over with 4 and kill Jace in the first damage step (gaining 5 life) and not strike in the second step or the optimal play would be to assign all 5 to the 1/1 killing it and getting 5 life and then hit Jace in the second step gaining another 5 life and killing him.