Can OTP still be broken if there are several messages being sent with only a slight variation in a password? If an attacker were able to get a hold of 2 encrypted messages that were both encrypted in a one time pad with a password that were each similar but had one change in the first character, could it still be cracked?
Let's say my first set of data was encrypted with "1password" and my second set of data was encrypted with "2password". Could it still be easily cracked?
What do you mean 'a one time pad with a password'?  One time pads don't take passwords, they take samples of truly random data as long as the message.

If what you're doing is taking the password, repeating it N times, and using that as if it were a random one-time pad, well, that can usually be broken even if you don't send a second message.

If what you're doing is you do have a truly random pad (which you are calling the password; however if you message is 10,000 characters long, so is your 'password'), and if you send a second message with a slight modification of the pad, then yes, it can be broken; the standard attacks against a 'two-time-pad' will work well (even if the pads are slightly different).

Bottom line: if you're relying on a 'one time pad', then it is mandatory that the pad be random, as long as the message, and not related to any other pad you may use.  If you violate this, the security proof for one time pad does not apply (and you are usually, in fact, insecure).