Strange problem related to DNS

My laptop, a Dell Inspiron 5520, connects to all networks without issue with one exception: my home network. Although I can connect it to the wi-fi, the connection has all kinds of strange issues. Programs that operate independently of DNS like Skype work without issue. Generally when I run a network diagnostic it will report that it cannot determine the problem, but occasionally it will report a DNS conflict. Web browsers will sometimes let me get though to a website, and if I do manage to make it onto a site I can navigate it more or less without issue. The majority of the time though the connection will fail. I've found that the best time to try to get though to a site is immediately after connecting to the wi-fi.

I'm convince the problem lies within the DNS, but I've done quite a few fixes and none have changed my situation. I tried manually setting the DNS of my computer's connection to that of my router, I tried using Google's DNS servers, I tried flushing the DNS cache, disabling all firewalls, and one or two other things the support team at my network recommended. Nothing has worked. I'm at wit's end here, because I'm clearly connected to the internet, and it works at times (albeit with a limited pool of websites), but there still seems to be a persisting DNS issue despite fixes that should logically solve the issue have been applied. Any help is greatly appreciated.