Hiya!
I've just finished this year's application cycle and been offered a place for HSPS so I can try and see if I can help you.
Firstly don't worry about finding the large workload tough - 5 AS is a lot of work, and I would advise you not to continue with all 5 next year, the application process takes up a lot of time and energy, and even with 4A2s it can be difficult to juggle everything.
Your primary concern should be your upcoming exams, as Cam place a heavy emphasis on AS UMS scores. Make sure you do as well as you can in these exams as they will be extremely important both for a Cam application and any other you will be applying for.
A levels can be tough, especially doing a number of essay subjects. Don't be put off by TSR people saying it will be too hard - it didn't put me off!
Work experience is definitely not necessary for this course - instead focus on your academic interest in the subject - things like essay competitions, public lectures, extra reading. If politics is the side of HSPS you are most interested in, I recommend the R.A. Butler Politics essay comp which is run by Trinity College, the college that offered me a place. I didn't enter it, but a girl I met at interview did, and although she didn't win, she was offered a place by Trin anyway, so do a bit of research for that.
However I do understand that work experience is something nice to have, especially if you want to go into career politics which it sounds like you might. My w/e was at Channel 4 News as I am interested in political journalism.
Your local MP is always a good start, as far as I know you need pretty good connections for the foreign office, but Cam won't be impressed by that anymore that grassroots campaigning, so if there are any local elections coming up see if you can volunteer for those. Remember that a large part of the politics course of HSPS is political thought, which is different to real-world politics, so don't get bogged down with practical experience - reading is the best way to broaden your knowledge of this aspect of the course. The suggested reading list I think should be on the HSPS faculty website, but otherwise give them an email.
PM me if you need any more help! Good luck