The truth is, yes, you probably should.
Is it as important as your resume? No, obviously your resume is the essential piece of jobseeker communication. The resume is where you should focus the majority of your efforts. But simply ignoring the cover letter isn’t a great idea.
Some postings specifically ask for a cover letter. In this case, sending your resume without a cover letter is directly ignoring the first instructions you will ever receive from your potential employer. This is not a great first look for you.
In some cases, job seekers have extenuating circumstances that can be a little tricky to address in the resume. For example, I’ve had many clients who are looking for a lower paying or less senior level position than they currently have for family reasons. While I wouldn’t recommend specifically stating that you’re looking for less responsibility (because, let’s face it – no one wants to hear that) you can state that at this point in your career you are looking for a change in direction. In some cases, it’s not a want, it’s a specific need that dictates this move. I’d advise you to address it concisely, without getting too personal. I never like to leave employers asking questions.
Do you need a new cover letter for each position?
Honestly, no… you probably don’t. You do need to customize it as much as possible with the company and the hiring manager’s address information, title, etc. If you don’t know the hiring manager’s name, it’s ok to say Dear Mr. or Mrs. Hiring Manager, but you’ll certainly be able to fill in the company’s name and address in the inside address block.
Cover letter’s need not be a novel and should under no circumstances extend beyond a page. Start with a bold statement if that’s comfortable, or express your sincere interest in the position. I like to include a few bullets that exemplify your performance and offer the hiring manager your confident assertion that you can improve performance for their team.
Close with sincerely or something traditional, and like the resume, if you’re printing it always use white or off-white resume paper. If you’re emailing it, it’s okay to copy and paste body of your cover letter into the email and attach the resume. Or reference both attachments, as in “Attached please find my resume and cover letter in response to your posting for Position Name on Job Board Name. I can be reached at Phone Number or Email and look forward to hearing from you soon.”
One final note…
Be sure to never mention salary expectations unless specifically requested. And when requested, be general, give a range (make sure your goal is in the bottom 1/3rd of that range) and don’t give reasons based on personal income. Your justifications, if requested, should be based on your value to the organization, not your financial obligations, which are unfortunately irrelevant to a hiring manager.