I'd like to offer recommendations where the existing site could possibly just have an SEO problem that won't go away just by getting the new domain value, however when I went to rocketproblems.com just now, I got a "server not found" message.

So without seeing your site, I'll offer insight on what to consider doing if you do the switch.

Exact match or partial match domains do offer value.  They are, however, on Google's radar for possibly getting less value than they do now, though Matt Cutts' exact language in a video earlier this year was they're considering a minor / slight reduction in value.

I wouldn't go by the competitors visible actions.  Just one month is not long enough to determine value of such a transfer.  It could easily take a couple months or longer to rebound and build the proper new trust factors that the new domain name will require, including links to the new domain.

Links to the existing site should be examined to find out which ones you can request being changed to the new domain name as well, since 301 redirects (vital to ensure these are implemented across the board and tested for verification) do take a hit on passing value.

I would also recommend a full press release through PRWeb.com or PRNewswire.com upon change to the new domain, as well as a social media campaign to promote the new domain.

So if you are prepared to wait out the potential drop over an extended period of time, do a full-court press.

I routinely do audits on sites with tens of thousands or hundreds of thousands of pages.  We've found good results simply by appending the page info to page Title & Description  so:

Designer Handbags | Unique Pocketbooks | Page 3

It's really a process of experimenting over time to find out the method that results in the most URLs indexed that in turn brings the most relevant traffic.  Personally I wouldn't have one for each category, yet without tests there's no conclusive reasoning either way.

To add to Corey's response, I'll repeat what I just provided another question here on Pro Q&A.  Sitemap.xml files can handle a maximum of 50,000 URLs, however I've seen them choke with as few as 10,000.  Its important to run them through a tool like tools.pingdom.com to ensure they load within just a couple seconds.

Then submit them through Google/Bing webmaster systems and then see if they succeed in crawling all of them.

bit.ly pro would work.  Personally I use yourls.org

Exactly. It's why I created the http://seoaud.it shortener for my own needs.

hahaha wLoudogg - don't even suggest that - I've been offered such nonsense plenty of times.  Never makes it any more worth the insanity

It's ultimately irrelevant from an SEO perspective - so it comes down to click tracking.  If you have different actions, you'd want a unique shortened URL for each action.

I'd also suggest considering setting up your own URL shortener, rather than relying on 3rd party.  As great as Bit.ly is, and as much as Twitter offers their own, the only way to ensure a short URL that's generated is always going to be around is to have your own.