I only thin the putty when I need to use it....I keep my tubes in a grip seal bag and then keep them in an airtight box....like a plastic lunchbox or similar, and so far never had the tubes dry out.
I have cellulose thinners at my work place, but you can get little bottles of cellulose thinners like this from Wilkinsons.
Wilko Cellulose Thinners 125ml | Brush Cleaners | | Preparation from Wilkinson Plus
I have worked with Cellulose thinners most of my working life, but please use caution with this stuff, the vapours can strip your lungs quicker than you may think, use in a well ventilated area....and dont use it for prolonged periods of time....One of the early signs of over exposure is believe it or not is an itchy sniffy nose.
Ray