General :: Sed Command - Find Line Not Beginning With Double Quote

I need to find each line in a file which does NOT begin with a double quote (") and append that line to the previous line. I have been successful doing this using the following command:
cat filname.csv | sed -e :a -e '$!Ns/
[^"]//;ta -e 'P;D' > newfilename.csv

My issue is the substitution. As you would expect after the line is appended to the previous line the first character is removed. I need it to not be removed. I tried:
cat filname.csv | sed -e :a -e '$!Ns/
[^"]/&/;ta -e 'P;D' > newfilename.csv
but it just hangs.

Goal:
Input:
"line 1"
line 2
Output with existing sed command is:
line 1ine2
I need it to be line1line2.