A few things to clarify - a theme with a "responsive" layout will automatically adjust to best display in the size screen a reader is reading it on, whereas a "fixed" layout will not, but this has to do only with the display of your site after your posts are published. This doesn't affect how things look when you compose them in the Dashboard versus how they appear when you publish. While you're welcome to upgrade, none of our upgrades necessarily change how images are displayed on your site.
What's mostly happening for you is that you're attempting to display images side-by-side by justifying some of them left and some right. This doesn't work, because the text wraps up between them. Rather, to display images side-by-side, you should justify them all either right or left, as described here:
However, that is a bit tricky, because you then have to have the images be the correct size to display on the same line, small enough not drop down to the next line, but large enough to take up the full line, because the text still wants to wrap around them to whichever side.
Instead, I recommend experimenting with justifying some of your images in the center (either single images, or a few arranged vertically) rather than to either side. Also, if you want to display a few images in a row, try inserting a gallery, which fixes them in a grid:
You don't have to put all of your post's images in a single gallery. You can insert a gallery of, say, four images underneath one paragraph, and three under another. But that will solve your problem of your images seeming to shift around on you.
Finally, try browsing some other travel sites on WordPress.com and seeing how those bloggers handle their images. That will give you some good ideas on how to display yours:
If you see something you like, but aren't sure how to achieve it, let me know - I'll be happy to help you out!
Sorry for the lengthy reply, but hopefully this gives you some good ideas for how to proceed. :)