Start rotating troops. Get around twenty out and put the entire lot on offensive; give them a week put them in and take another out, etc.
That's the great part about being the attacker - you can rotate troops without having to move around, and besides, it seems that you're grinding the German army to dust at least. Keep attacking, and every division destroyed will be a godsend.
What Hyo said. Note that both you and the Germans have several divisions with next-to-no organization. For the attacker, they will simply stop attacking when they reach zero organization, and you can send them back into the fray in a few days. For the defender, they retreat, and are out of action until they both finish retreating into another province, and can rest up from the defense.
Besides, once you have more divisions than your commander can handle, the excess divisions stop being effective. The AI doesn't seem to realize that, so you'll likely note that many of the German's later divisions entering the fight aren't really contributing much.
Armored, motorized, or mechanized assets would be ideal for subsequent waves of attack. But use whatever isn't dug in elsewhere.
EDIT for newest post: Don't feel bad if your offensive fails. It happens. Just as long as you hurt them more then they hurt you, you're doing well.
And remember where the Germans were at this point in time historically: Drinking Pere in Paris. You're doing well.
Oh, and I'd suggest reading
this LP by Fishbreath for inspiration and ideas. Granted, it's Arsenal of Democracy instead of Darkest Hour, but from what I can tell, it's quite similar. Fishbreath is whipping the Germans good, despite many, many failed offensives.