Most pain killers and sleeping pills make me sort of high. Does that mean I will get addicted quickly?

No. It means your body is sensitive to them............Before getting addicted you will have other side effects so take it as needed and at a lower dose than normal.
Not Necessarily. Many patients feel the sedating and/or euphoric side effects of narcotics and benzodiazepines. Most patients will see these side effects as negatives, not positives. If you find yourself looking to reproduce that feeling, or you already have background addictive behavior, you may well be at risk. If concerned, discuss the matter openly with prescribing physician. Help is available.