We can check 404 error using Google webmaster also. So, would it be more beneficial to check server states and logs?
It may take longer for Google to report 404 than looking at your logs, as Google must crawl your URL in order to get 404 response, and then there is a furthe delay until the data is reported in WMT. Furthermore, not all 404 are reported in WMT, there may be only a selection/subset.
Checking logs as advised by aristotle is better as you get more accurate info faster.
If you have a sitemap from the old site, you may consider using a tool such as Screaming Frog to crawl all URLs from the old sitemap to ensure they are redirecting correctly.
Also - what Lucy above said is important - when you are looking at the traffic in Google Analytics, how is your data reported? Are you using the same GA code and therefore have both site's traffic combined? Or do you have a separate GA code for the old and new site, in which case the traffic to old site should stop and the traffic to new site start to appear?
If combined traffic has decreased for almost 50%, could you analyse which pages / which site sections have lost traffic and from which sources?
If you go to Google Webmaste Tools (Search Console), and look at your Queries, what happened with your impressions, CTR and average ranking position - with 50% of traffic gone you should see a visible change there.