A prime lens vs a zoom lens for a holiday/landscape?

Hey guys. I know this MAY be a bit vague to answer but i'd appreciate ANY input. I currently own a D5100 with a Kit 18-55mm lens (which is worse than my old Bridge camera lens in terms of quality. I think it has a fault) and a 55-300mm lens which is amazing. Seeing as I normally use the 55-300mm lens I have never thought about getting a wider lens… Until now.

In a few weeks time I will be travelling to Greece with my girlfriend on our first holiday. As it's our first holiday I want to be able to get some great photos of us and of the scenery (beaches, etc). As you can tell my 55-300mm lens isn't really going to do the job. So, I've decided to buy a new, somewhat cheap, lens that would be great for landscape/scenery, museums and also for portraits.

My budget is £100-£250 (at a push for a prime) but I will go a little higher for a zoom.

My first idea was to go for a 17-70mm lens like this one http://www.amazon.co.uk/Sigma-17-70mm-2-8-4-HSM-Nikon/dp/B00AXZZ034/ref=cm_cd_al_qh_dp_t

However, i've noticed a lot of people saying you can't go wrong with a good 35mm lens when it comes to Landscape and portraits photography due to the pin sharpness compared to a zoom lens. Is this true?

My only requirements is if possible it has to have some kind of Optical Stabilizer, fits to a D5100 and also is as cheap as possible. So if you're kind enough to give a suggestion. No £900 lens lol:) Thank you!

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