I live outside Cardiff (in Caerphilly) and work in Cardiff, and I don't think the rental market is nearly as difficult as some areas. It sounds as if the newness of the area and a new system combined with the stress of moving is getting to you. There are some cities where it all disappears very fast, but I don't think it's as big a worry here. Keep at it and you'll find something. What kind of property are you looking for? Do you have special requirements? Really, don't stress. There are good places available, and you shouldn't have too much trouble once you get the hang of how it works. Finding a new place is never fun, but you'll be able to do it okay. If you have a bit saved for deposits, rent, any agency fees, etc, you should have no trouble. If you are using housing benefits (from DH since you don't qualify at this point--assuming you're coming on a spousal visa) it's a bit more difficult to find people who are willing, but you totally still can find them.
Estate agents are different here than real estate agents in the US. It's really more like a management office that you might see in the US. They will have a slew of properties that they are paid to take care of and collect rent on, so of course they will only show you theirs as they won't make any money otherwise. Generally the landlord will then pay them a portion of the rent. Many also have an up-front fee.
Rightmove is a good place to start as it will have places from a variety of sources. If you have questions about different areas in Cardiff, let me know and I can help you out in knowing some of the nicer spots versus those to maybe avoid. It isn't always clear-cut as it depends on what kind of things you are looking for, and sometimes there are very posh areas just blocks from areas that are lower-income and have some troubles, but if you have questions about specific flats or areas, I'm glad to help out if needed. Of course, I just remembered you are moving with DH's family, so they should know the area too.
Good luck, let us know how you're getting on. I hope you come to love South Wales as I have. It will help if it stops raining enough for you to get out and enjoy this beautiful countryside. There is so much that is fun around Cardiff!