Is your paper wallet a private-key (weird big string) or a backup seed (list of words)?
To clarify your private key can look like any of the following:
Base58 uncompressed private key (note the first character)
5Kb8kLf9zgWQnogidDA76MzPL6TsZZY36hWXMssSzNydYXYB9KF
Can be imported in any wallet that has the import option
Base58 compressed private key (note the first character it can also be K)
L53fCHmQhbNp1B4JipfBtfeHZH7cAibzG9oK19XfiFzxHgAkz6JK
Can be imported in any wallet that has the import option
Hexadecimal private key (note the length is 64 and characters are 0-9 and a-f)
E9873D79C6D87DC0FB6A5778633389F4453213303DA61F20BD67FC233AA33262
Has to be converted to a base58 (WIF) private key like the 2 examples above before the wallets accept it. To do that you could use this open source tool:
https://github.com/pointbiz/bitaddress.org (run offline)
Base58 encrypted private key using BIP38 (note the first 2 characters)
6PRW8vhrsbZJdZDtVCKtfnPoN7ecWKraUagCDg7M9APkNGMxx3fbwMkysT
In this case you also must have the password that were used to encrypt this key and some wallets don't accept this type so it has to be decrypted first before you can import it in a wallet. You can use the bitaddress.org tool to decrypt it.
Password used here is "123" in case you want to test decrypting in that tool.
So, it I import both together, I will have access to the balance?
You do not need to import either your addresses or your public keys. The only thing you need to import is your private keys - Electrum will derive the public key and the address automatically from the private key.
Okay, I have two paper wallets and each one has a public and private key. Does the fact I have both unlock the wallet?
The wallet will be "unlocked" simply by possession of the private key. The public key and/or address are simply printed so you can look up your balance or deposit more coins to the paper wallet without having to expose your private key. The private key is all that is needed to spend the funds, and is the only thing you need to import to Electrum.