However, whether you do it as y= -(ln|x+2|- 3) or y= -ln|x+2|+ 3, the graph is 3 places higher than the graph of y= -3ln|x+2|, not lower!
For example, for x= -1, ln|-1+2|= ln|1|= 0 so y= -(0- 3)= 3 while -ln|x+2|+ 3= -ln|1|+ 3= 3. The graph y= -ln(x+2) passes through the point (-1, 0) while the graph of y= -ln|x+2|+ 3 passes through the point (-1, 3), three places higher.
If your problem is actually y= -(ln|x+2|+ 3)= -ln|x+2|- 3, then the graph passes through (-1, -3), three places lower.