Ytukyg

If when using a Vignere cipher I replace the key word often with a word earlier in the ciphered plain text would this be stronger due to not being able to do analysis with a repeated key or would it be fundamentally weaker due to sending the key for the next message for everyone to see? How long do the messages need to be in terms of key length to make this method stronger?

For example:
The key word orange has been shared with 2 parties. The message "my password is bob and the next key word will be science" is encrypted using the word orange and sent to the second party. They then send me back a message using the key word science to encode their message as well as sending me a new key word.

This system does not add security, only weakens it: the first message can be solved in a standard way, being a Viginère cipher. For this we only need that the key is quite a bit shorter than the message we encrypt, and statistics will do the rest. After that you know from the solution what the next key will be so the attacker has a simpler time. Stereotypical beginnings are a weakness anyway, and I fail to see what "my password is bob" adds, if we are already using "orange" as key? Also use random strings, not dictionary words, as keys.