A few old fashioned expressions:

Zut
Flûte
Crotte

I would suggest

pauvre de moi

as it include the "self complaint" component of "dear ME" (also poor me in English). Dear me is however also used in other context ‘sympathy, surprise’ hat are then different from "poor me". In which case I suggest (to remain polite):

Mince alors, zut alors

A translation depends on the context.

I Surprise

• Vraiment ?, Pas possible !, Ça alors !, Bigre !, Fichtre !, Ciel !, Mon Dieu !

• — This car can reach a speed of 490 km/h.
  — Dear me!
  →
  — Cette voiture peut atteindre la vitesse de 490 km/h.
  — Vraiment ?/Pas possible !/Ça alors !

II Irritation

• Mince, Zut,
• Purée, Punaise (colloquial) (a bit naive, I’d say)
• Merde (very colloquial to vulgar)

Dear me, I forgot to mail it. → Mince, j’ai oublié de le mettre à la poste.

III Sympathy

• Oh là là, Oh mon Dieu, Mon Dieu !

Dear me, the poor man must be suffering. → Oh là là ! Le pauvre homme doit souffrir !

IV Distress

• Oh là là !

All this work I have to do! Dear me, I’ll never manage to get it done… → Tout ce travail que j’ai à faire ! Oh là là ! Je n’y arriverai jamais …

https://www.wordreference.com/enfr/dear%20me
https://www.francaisfacile.com/exercices/exercice-francais-2/exercice-francais-57394.php