Yes, you can buy a bluewater capable boat of the size you're looking at in the price range you can afford. However, almost none of the bluewater boats you're looking at are going to have a "full bed" as a general rule. To have a full bed, rather than a v-berth or some other type of double berth really requires more beam than most of the bluewater-type boats are going to have in that size.
Hallberg Rassy 35, Southern Cross 35, Alberg 37, Van De Stadt designs, Cape Dory 36, and others are all within your price range and are proven bluewater capable designs.
However, if you are on a budget, and I seriously recommend that you reserve at least 15-20% of your budget for refitting, upgrading and modifying the boat that you do get, you may have to make some compromises.
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You know what the first rule of sailing is? ...Love. You can learn all the math in the 'verse, but you take
a boat to the sea you don't love, she'll shake you off just as sure as the turning of the worlds. Love keeps
her going when she oughta fall down, tells you she's hurting 'fore she keens. Makes her a home.
—Cpt. Mal Reynolds, Serenity (edited)
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Still—DON'T READ THAT POST AGAIN.