Addiction means continued use in spite of negative consequences. In this sense Benadryl addiction is not known to be a common problem. Since it can cause weird reactions at high doses I guess that is possible. People who abuse downers are usually trying to be somewhat emotionally “numb” – so that’s a possibility.
Many people confuse physical dependence with addiction. This results from taking something long enough or in high enough doses that you become physiologically adapted to it – then if you just stop it you can have withdrawal symptoms. In the latter case you just need to taper it slowly – it could take a few weeks, depending on how long you have taken it regularly.
The other issue would be what were you taking it for? Is there a condition that needs to be treated?