#Java, 655 bytes

public class M{public static void main(String[]P){String S="swallowed",T="There was an old lady who "+S+" a |.",e="I don't know why she "+S+" that fly,\n";String[]a={"fly","spider","bird","cat","dog","horse"};String[]q={"","That wriggled and iggled and jiggled inside her.","How absurd |","Imagine that |","What a hog |",null};for(int i=0;i<a.length;i++){p(T.replace("|",a[i]));if(q[i]==null){p("She died of course.");break;}if(!q[i].isEmpty()){p((q[i].replace("|","to swallow a "+a[i])).trim());for(int j=i;j>0;j--){p("She swallowed the "+a[j]+" to catch the "+a[j-1]+",");}}p(e+"Perhaps she'll die.\n");}}static void p(String v){System.out.println(v);}}

This is a golfed version of this answer. There's over 100 bytes saved on minor things.

Java, 655 bytes

public class M{public static void main(String[]P){String S="swallowed",T="There was an old lady who "+S+" a |.",e="I don't know why she "+S+" that fly,\n";String[]a={"fly","spider","bird","cat","dog","horse"};String[]q={"","That wriggled and iggled and jiggled inside her.","How absurd |","Imagine that |","What a hog |",null};for(int i=0;i<a.length;i++){p(T.replace("|",a[i]));if(q[i]==null){p("She died of course.");break;}if(!q[i].isEmpty()){p((q[i].replace("|","to swallow a "+a[i])).trim());for(int j=i;j>0;j--){p("She swallowed the "+a[j]+" to catch the "+a[j-1]+",");}}p(e+"Perhaps she'll die.\n");}}static void p(String v){System.out.println(v);}}

This is a golfed version of this answer. There's over 100 bytes saved on minor things.