I think that ultimately it is up to you to do whatever you feel like is safer for you. Given a choice nobody, be it here on TSE or elsewhere, will (want to) make this decision for you. What I can do however is to give you a few pointers to help you make an informed decision.
Sinking Might Be Hard, Drowning Not so Much
First things first the Dead Sea is denser than fresh water and the sea water you might be used to, so objects will float easier than in the other two scenarios. This however also means that objects will have a stronger force acting on them pushing them towards the surface. It follows that putting your head underwater will be harder than what you are used to in less salty water. This however does not mean that you cannot, or won't, drown. It just means that sinking is harder.
Drowning indeed can happen because once turned stomach-down the head will be pushed in the water as the legs are pushed to the surface. Moreover turning back to stomach-up will be harder than in less salty water. This post on io9 puts it in simple, yet effective, words:
Floating on their backs, a person's face is lifted well clear of the water. If they turn over, or if they trip on their way into the sea and fall face down, then the entire back of their body is clear, but their face is pushed into the sea. Anyone who has been in a pool knows that, in regular water, the easiest way to lift the head up and out of the water is to force the feet and lower body down. Easy in fresh water, or in the relatively saltless ocean. Harder when the water keeps forcing every part of the body up. Because the water is dense, it's hard for a person to push an arm into it and turn their body over.
Still Thinking of Swimming?
Arguably the best piece of advice one can give is for you to consider your swimming skills. Are you a strong swimmer? Are you confident in your ability to spin around to a safer position when in trouble? Are you used to not panicking in and under the water? Once again only you can answer these questions. It is worthy to note that overconfidence in and around water tends to be quite dangerous. If you want to try this please do it with someone you deem capable of getting you out of whatever dangerous situation might arise supervising you closely.
Finally here is a quote from the Dead Sea WikiTravel wepage:
Do
[...]
- Beware! Several people drown every year in the Dead Sea because they do not obey the rule: Only float on your back. Accidents happen when someone tries to swim normally (stomach first) in the water - the legs will float better than usual and the head will be submerged. Note that this applies to weaker swimmers, and specifically to attempts to swim breaststroke. Breaststroke is also made difficult by the fact that the legs are raised too high in the water to provide normal forward motion when kicking. Moreover, the salt in the water stings cuts and causes great pain if it comes in contact with the eyes, adding to the panic if one's head is under water. A strong swimmer can easily swim freestyle; if you plan to try this, goggles are essential and should be tightly fitted. Although safe for a strong swimmer, and an unusual sensation because of the buoyancy of the water, it is not an undertaking most people are likely to sustain for long. Even with the eyes protected by goggles, water will get into the nose and sting, and onto the lips and inevitably into the mouth. It tastes disgusting.
