This is a very common and understandable problem because the communication really is and has to be largely about you, the applicant. Still, here are two tactics for your consideration:
One, shift a lot of the content out of the cover letter and into the résumé, where grammatical predicates are routinely rendered with their grammatical subject, “I,” understood rather than expressed. This will likely make for a very short cover letter; so much the better. The old way was to get up one résumé, professionally formatted, for all potential jobs, and then tailor the cover letters to each prospective employer’s desiderata. Today, with high-powered formatting and typesetting at all of our fingertips, you can tailor the résumé itself to each separate application.
Two, try launching some sentences using the pronoun “you,” meaning the specific potential employer. If you hire me, you can expect . . . you can draw upon these skills and this experience for these your purposes. This tactic will lead you to look at the proposed hire from the employer’s point of view, which can lead to manifold improvements in your rhetorical approach, not only in the résumé and cover letter but also in the interview.