The problem is what WordPress does with the code. It seems to convert it to a different format (from embed tag to iframe tag) and in the process changes parts of it which seems to break it.

Here's the relevant badge code portion as an example. The way it's provided by SmugMug:
Code:
http://cdn.smugmug.com/swfs/badge/flashbadge.swf?introMode=fadein&autorepeat=true&gridRows=3&gridColumns=4&gridSpacing=1&gridDelay=2&autoHideDelay=4&slideshowDelay=4&autostart=true&aboutlink=&sharelink=&background=%23000000&displayMode=full&useLargeImages=true&albumID=19970112&albumKey=vJ8Rk9&order=norandom&width=200&feedType=&nickName=melissagainey&height=200&BadgeHost=cdn.smugmug.com&showCaptions=false

and the same portion from your WordPress blog (decoded for easier readability):
Code:
http://cdn.smugmug.com/swfs/badge/flashbadge.swf?useLargeImages=true&nickName=melissagainey.smugmug.com&feedType=&BadgeHost=cdn.smugmug.com&AlbumID=19970112&AlbumKey=vJ8Rk9&ImageID=1559686576&ImageKey=dTDSdDx&width=300&height=300&gridSpacing=1&gridColumns=4&gridRows=4&background=%23000000&preloaderColor=&preloaderGlow=&gridDelay=1&slideshowDelay=4&order=norandom&forceSize=&showCaptions=false&introMode=fadein&aboutlink=&sharelink=&albums=

Apart from that stuff has been juggled around, you'll note that WordPress changed the parameter names (which are cAsE-sensitive).
albumID was changed to AlbumID
albumKey was changed to AlbumKey

I suggest that you report this issue with WordPress to see if they can correct their plugin that does the conversion.

I'm not sure if you can directly edit the code yourself, but if you can, try to correct the capitalization issues that WordPress is introducing to match the parameter names to what there is provided by SmugMug.

If you can manually edit the code, DO NOT copy and paste the above code from SmugMug as it won't work in its current format. You would need to manually change the individual letters that are incorrect.
__________________
Sebastian
SmugMug Support Hero

I changed the capitalizations and it still didn't have en effect. I think it is something that is happening with WordPress behind the scenes...

Has anyone else run into this issue?

[gigya src="http://cdn.smugmug.com/swfs/badge/flashbadge.swf?useLargeImages=true&nickName=melissagainey&feedType=&BadgeHost=cdn.smugmug.com&albumID=19970112&albumKey=vJ8Rk9&width=300&height=300&gridSpacing=1&gridColumns=4&gridRows=4&background=%23000000&preloaderColor=&preloaderGlow=&gridDelay=1&slideshowDelay=4&order=norandom&forceSize=&showCaptions=false&introMode=dropdown&aboutlink=&sharelink=&albums=" width="300" height="300" wmode="transparent" allowScriptAccess="always" ]

and the format and functionality appear to be working great. However, they aren't my pictures. The link to my pictures is http://cdn.smugmug.com/swfs/badge/flashbadge.swf?useLargeImages=true&nickName=melissagainey&feedType=&BadgeHost=cdn.smugmug.com&albumID=19970112&albumKey=vJ8Rk9&width=300&height=300&gridSpacing=1&gridColumns=4&gridRows=4&background=%23000000&preloaderColor=&preloaderGlow=&gridDelay=1&slideshowDelay=4&order=norandom&forceSize=&showCaptions=false&introMode=dropdown&aboutlink=&sharelink=&albums=

When I look at the two side by side they look the same to me, but I am clearly missing something...

The blog post is here: http://melissagainey.wordpress.com/2012/07/28/disneys-wild-africa-trek/

Thank you again! As I mentioned earlier, I'm new to this and appreciate any and all help!‎

I am unable to post the following code into my wordpress.com post properly (Copied directly from Smugmug's showoff badge:

'object classid="clsid:D27CDB6E-AE6D-11cf-96B8-444553540000" width="300" height="300"><param name="movie" value="http://cdn.smugmug.com/swfs/badge/flashbadge.swf?useLargeImages=true&nickName=melissagainey&feedType=&BadgeHost=cdn.smugmug.com&albumID=19970112&albumKey=vJ8Rk9&width=300&height=300&gridSpacing=1&gridColumns=4&gridRows=4&background=%23000000&preloaderColor=&preloaderGlow=&gridDelay=1&slideshowDelay=4&order=norandom&forceSize=&showCaptions=false&introMode=dropdown&aboutlink=&sharelink=&albums=" /><param name="wmode" value="transparent" /><param name="allowScriptAccess" value="always" /><embed src="http://cdn.smugmug.com/swfs/badge/flashbadge.swf?useLargeImages=true&nickName=melissagainey&feedType=&BadgeHost=cdn.smugmug.com&albumID=19970112&albumKey=vJ8Rk9&width=300&height=300&gridSpacing=1&gridColumns=4&gridRows=4&background=%23000000&preloaderColor=&preloaderGlow=&gridDelay=1&slideshowDelay=4&order=norandom&forceSize=&showCaptions=false&introMode=dropdown&aboutlink=&sharelink=&albums=" width="300" height="300" wmode="transparent" allowScriptAccess="always" ></embed></object

I stripped it down to this thinking this would work:

'[gigya src="http://cdn.smugmug.com/swfs/badge/flashbadge.swf?useLargeImages=true&nickName=melissagainey&feedType=&BadgeHost=cdn.smugmug.com&albumID=19970112&albumKey=vJ8Rk9&width=300&height=300&gridSpacing=1&gridColumns=4&gridRows=4&background=%23000000&preloaderColor=&preloaderGlow=&gridDelay=1&slideshowDelay=4&order=norandom&forceSize=&showCaptions=false&introMode=dropdown&aboutlink=&sharelink=&albums=" width="300" height="300" wmode="transparent" allowScriptAccess="always"]

However when I do this it pulls pictures from someone else's gallery? I know I'm doing something wrong and I'm new to this. I'd appreciate any and all help.

Thank you!
Blog url: http://melissagainey.wordpress.com/