Zsh + coreutils, ⁴⁰⁶ 375 bytes

g()ls -t<>"She$d the $1 to catch the $2,"
set bird spider \ swallow iggled\ 
<<Z
${b=There was an old lady who${d=$3ed} a }fly.${5=
I don't know why she$d that fly,
Perhaps she'll die.

$b}$2,
That wr$4and $4and j$4inside her.
`g $2 fly`$5$1,
How absurd${6= to$3 a }$1.
`g $@`$5cat,
Imagine that$6cat.
`g cat $1`$5dog,
What a hog$6dog.
`g dog cat`$5horse,
She died of course.

Try it online!

Tricks:

• the g function creates the line as a file and then ls lists the files in the current directory in order of creation (-t), to avoid having to repeat the previous lines every time.
  • I used <> (which creates the file before the command is executed) instead of >...; because using multiple commands in a function requires {} around them
• set bird ... assigns the words to $1, $2, etc.
• \ swallow includes a space at the start of without needing to use quotes
• <<Z starts a heredoc which is effectively a long interpolated string
• ${b=There...} is a shorthand form setting b to that string and substituting its value in
• Indexed variables like $1, $2, etc. can be substituted without a space afterwards (rather than being interpreted as a variable called 5dog, for example), so we prefer them as it allows us to include the space inside the assignment to avoid repeating it
• g $@: since we set $1 and $2 to bird and spider respectively, we can pass all the numbered variables to g in one go with $@ instead of g $1 $2 (it ignores the extra ones, $3 etc.)