Career transition help

I want to keep this short and simple. I took the GMAT and I have got a good score. So now I want to focus on my work experience and extracurricular till I apply somewhere around 2025. Current work experience - 1 year as a research analyst at an international consumer media MNC Aim - I genuinely aspire to work in the client servicing department of advertising. Client facing roles are a big green flag to bschool adcoms and I am genuinely interested in that field (I have grown up watching & rewatching Mad Men) Issue - My current work experience is barely similar to my goal. The company I work at is a big brand name and I have some transferable skills that I can put in my resume but getting my foot in the door seems difficult. I am looking for people who can help me out either in terms of tailor made guidance or in the form of referrals. I have stalked the HRs of different agencies like Ogilvy, Denstu Creative, Publicis Groupe, VML and MullenLoweLintas to send them an e-mail making my case and highlighting my transferable skills over the past 2 weeks but I have no leads whatsoever. This must mean I am doing something wrong and I wish to fix it before I try my luck at Schbang. Thank you!!