To be clear, "I haven't much experience" is perfectly fine, and in fact it's a very common pattern in conversation *outside the US*:
- I haven't the foggiest notion
- We haven't much money
- I haven't the time
- You haven't any idea == You have no idea
As Tricia says, in US English we tend to negate the verb instead of the adjective:
- We don't have much money
- I don't have the time
- You don't have any idea == You have no idea
... but both constructions are grammatically correct.
I hope this helps.
¡Saludos!