# we have rejected secret and shameful ways

This means that Paul and his coworkers refuse to do "secret and shameful" things. 

# secret and shameful ways

"the things that people do secretly because they cause shame" 

# walk in craftiness

"live by deceiving people" 

# we do not mishandle the word of God

"we handle God's message correctly" or "we use the word of God properly" 

# we recommend ourselves to everyone's conscience

This means that they provide enough evidence for each person who hears them to decide whether they are right or wrong.

# in the sight of God

"before God" or "with God as witness"