There is an unspoken assumption thro' the hobby that refractometers are 'better' than hydrometers as they are more accurate - this is not always true.
For sure swing arm hydrometers are rubbish. Even with frequent cleaning to stop weighting of the arm, and sticking, theyare still hopeless and to be avoided.
The point to bear in mind is that of hydrometers, refractometers and even digital meters NONE MEASURES SALINITY DIRECTLY. All use a physical phenomena (in order density, refractive index, electrical resistivity) that can be linearly correlated to salinity, and then back out the figures from the correlation to give a salinity. However all these 'salinities' are an approximation due to accuracy of the tool and also all of these phenomena are affected by other phenomena, principally temperature, but for the digital readings also the exact seawater composition can make a difference.
From a practical point of view, what to do, what to buy. For a start, if you have a floating hydrometer don't necessarily thro' it away. I have a lab grade hydrometer (froim Brennan in the UK, redistributed by TMC, not available in the US tho' the equivalent is somewhere) that comes with a +- accuracy reading. This is very good, reliable, somewhat easy to use. Your floater might be as good as this , might not - you need to investigate by looking at it on the scale , or on the packaging. For the same price as my hydrometer I could get a refractometer from Marine Depot, STM or whoever for 30 to 50 euros/dollars. This is easier to use, and as lorna has worked out, has calibrated it's zero to distilled water successfully. To check it's accuracy at a real salinity you would need to compare it to a stock solution of 35 ppm salt mix. Normally at this price point the +- accuracy is either not defined, it is a little poorer than my hydrometers reliability, repeatability, but it is easier to use. To beat my hydrometer you will need to spend 200 dollars plus to get a more reliable calibrated refractometer, and you will also need to spend more time calibrating it.
For me the digital meters are not worth it. they need repeated calibration and careful cleaning. If you are using one as a computer controller then they are the only thing you can use, but they do require much maintenance then, wear out quickly and are still not 100% accurate for a good number of reasons.
For me most hobbyists can use either a refractometer or a decent floater. The differences are negligible