Oh yeah, I don't think you've truly experienced full throttle stress until you've been in a situation like this! I've gone through this way more times than I'd like to remember, but here's some of the tricks I've picked up and developed along the way.
First off, start small. Pull all of the changes together to form one *current for the moment* script. This may take a few hours and some heavy cutting and pasting, but you will be much more focused once you have one base copy done. This process is always made easier if you have an electronic version of the script. If you don't have one, see if the playwright does. Otherwise get out the scissors and glue!
When doing up this version, leave extra space. Widen your margins a bit, space lines at 1.5 instead of single spacing. What this does is leave room for write ins during the rehearsal process.
Once done, consider this your base. It should be the cleanest, most up-to-date copy you could have at that moment. Now here's where most non-SM people cringe - Copy it and send it out to everyone. From that point on, when changes are made everyone has the same base to work off of (this logic works great on penny pinching PM's
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As changes come there are many different techniques to keeping everyone updated. I keep a notepad just for line changes that happen in rehearsal beside me. I write the change in my book, then jot down the page number and note how many changes there were on that page. At the end of rehearsal I have a chart for the frequent smallish line changes/tweaks. Something like:
Pg Original Line Change
14 The prisoners are free. The hardened criminals escaped!
15 Wowezowie - I couldn't believe... The speech is cut
I usually do up this chart in Excel and that way, common phrases pop up instead of me typing hundreds of 'the speech is cut.' This chart is great for e-mailing out to production people, and printing off hard copies for everyone to put the changes in their base scripts. For the actors call board, I print each day off in a different colour and post it.
For larger scale changes (ie. Scene 2 looks like the lines were put in a blender and set to frappe) if you have an electronic version you can go in and make those changes pretty easily and then copy and send out the new pages. Once again, for the actors call board I do post them on coloured paper. Make sure that you put the date on the new pages! If there are that many changes happening, this will be a life saver in making sure everyone is as updated correctly as possible. If you've had to cut and paste your script together because you don't have an e-version of it, type up the new scene.
This is where that base e-version of the show could come in very handy. Keep updating it regularly. You or an ASM can go in and update and save it as different versions so that if they decide a few weeks down the road to go back to what they had - you're set. This is also great because at some point you may have to provide someone with a new/replacement script and you'd be set with an updated one on file.
It is tough and a lot of work when you do a show that is constantly changing. The best you can do is know that this is happening and be prepared for it. I'm sure you're going to get a lot of tips here, look them over and put together something that works for you and your company. And best of luck with it all, and remember, you're not alone in this!