I was hoping to get some additional information from you, but for the moment I can give you the following information to try to open your safe: First, I recommend replacing the batteries again, and make sure they are alkaline or lithium for electronics--NO rechargeable batteries. Make sure there is no corrosion (rust, leaked battery acid) in the battery holder, and if you find any, gently remove with a Q-tip and rubbing alcohol. Be sure battery 'poles' are aligned properly in the holder--negative to negative and positive to positive.
Leave no more than 5 seconds in between entering numbers of the combination on the keypad. Make sure if the key is round and hollow, and it's made to release the cylinder or plunger inside the keyhole, the plunger is in the 'out' position; if it is pushed in and the key won't release it, the safe will not open. Also, if you can eventually release the plunger, REMOVE the key from the lock before entering the combination.
If the handle won't move at all (up or down), it may be stuck or defective. Try the following: spray a little WD 40 around the handle where it meets the door and try to loosen it up, gently. If it won't move, don't force it. Move the safe onto it's sides and lean it forward and toward the back, so if something inside is blocking the opening mechanism, it will move away.
Take a rubber mallet or a hammer with a towel wrapped multiple times, around the head and strike the door above and below the keypad, then all around the edges of the door. Safes with rubber gaskets tend to swell, preventing the door from opening. Also, push against the upper left corner (as you face the safe), as if you were closing it, as this particular part of the door tends to get stuck the most. Also, hit it there with the mallet/hammer.
These instructions are mainly for Sentry safes; if yours is by a different company, please let me know and I will research further.
I hope you can get your safe open!
Please rate my answer positively so I can receive credit from the site for my time and assistance. Please do not rate negatively if you still can't open the safe, because that is out of my control. As a last resort, you can call a local locksmith or safe technician or call the manufacturer of your safe to troubleshoot, further. If you need any contact information for the safe company, please ask me.