Don't let your budget determine your strategy. I'm not a fan of spending money because you have it. I would focus on making sure that the clicks you are getting are actually turning into paying customers for the business and once you can confirm that the clicks are actually working for you, then you can scale up your campaign. But it can be done. I have a client campaign running that I spend $15,000 per month on and still maintain a >4.5% CTR. So don't let your budget decide your strategy, let what's working decide and scale it upward.
Giving specific advice on your campaign is tough because we don't know if it's a local campaign, product campaign, affiliate campaign, etc.
You asked, "And should each campaign have its own set of unique keywords? Should these keywords never appear in more than 1 campaign?"
Quick answer: 1. Campaigns should have adgroups that target their own niche and then those adgroups should have their own ads that target specific set of keywords. 2. I personally feel that if you do #1 right then keywords will not appear in multiple adgroups.
I like to build adgroups around 1 product and then ads based on keywords that are intuitive for that search and then I target each of those keywords with their own CPC. For instance, if I'm selling widgets I'll set up a campaign called widgets, an adgroup called blue widgets, and then have keywords like blue widgets, buy blue-green widgets, light blue widgets. I'll then try to target each of those keywords with their own CPC instead of setting the at the adgroup level. I also try to make sure that all my keywords in my ad groups are "phrase match" instead of broad or exact. And, like Sam mentioned, I negative keyword out keywords that don't relate to my strategy. For instance if you only sell new items, you could add the negative keywords -refurbished, -old, etc.
I don't use PPC like most pros however, I try to only target people who need "Now" things and keep the competitive shoppers to the organic side. This will generally help keep your cost down and conversions up. There are some tricks and things you can do in the title and text to help get only now shoppers. Also, enhanced campaigns in AdWords now get higher Quality Scores so I would definitely look into optimizing that along with each landing page you have on your own site.
Here is some information on enhanced campaigns from Google.
I'll repeat myself again, but let your goals decide what your strategy is and not your "lack of budget spending". I generally tell clients that I need $10,000 per month on a campaign but only spend roughly $7,000 or so. As your quality score and CTR increases, your CPC will go down. If you worked for me and weren't spending all of the budget I gave you and we were still meeting our goals, you'd be getting a bonus.
Hope that helps.