What do I need to do for HTTPS switch in Webmaster Tools?
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My site is currently verified using a meta tag for both Google and Bing. Will I need to recreate the meta tag or will I be able to use the same one?
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Currently there is not option in GWT to change the site URL to HTTPS. This was the known issue.
For now, if you decided to switch from http to https, properly redirect using 301 redirect with right Canonicalization.
Hope this helps. Thanks.
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I have that part down but in the instructions to move the site Google states -
Likewise, if you verify ownership with an include file that references meta tag or uses Google Analytics to verify ownership, ensure the new CMS copy includes these as well.
It seems like I can keep the same meta tag, but I am not sure.
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Do this step,
Feel confident by performing these steps

Go to "Add or Remove owners" then select "Show Verification Methods" and now hit "Verify" with present meta tag. I assume it should show "Congratulations..". If not, use google analytics to verify.
Actually, you no need to verify it again. You will get the option soon at "Change of Address".
Here is the good read Google Change Of Address Tool Does Not Support HTTPS Migration
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Well that says the change of address tool isn't working so that doesn't make me feel too great.