Try playing the animation only once when starting to move, using a variable. Something like:
var should_play_anim = true
func _fixed_process(delta):
if not Input.is_action_pressed("move_forwards"):
get_node("AnimationPlayer").stop()
should_play_anim = true
if Input.is_action_pressed("move_forwards") and should_play_anim:
get_node("AnimationPlayer").play("walk")
should_play_anim = false
(Note that this code has not been tested.)