Sorcerer claws are a cool concept for a non-standard build. When I first saw them, I immediately wanted to build a Sorcerer/Eldritch Knight/Dragon Disciple gish who wades into battle and rends people with his claws.
With the rules as written, that probably only works for an NPC who fights one battle and dies (or escapes and fights no more that day). For a PC, there are not enough rounds to have them available for every combat. Instead, when you get grappled or run out of spells, the PC can look fearsome while he dies instead of looking helpless. I suppose that is something.
The potential problem with just extending the duration by a lot is it encourages a one level dip. If one level of sorcerer lets you have claws whenever you are in combat, that plus access to shield and true strike probably mean many optimized melee builds include one level of sorcerer. With claws and a reach weapon, you would threaten 5' and 10'. A full attack lets you do 2d4 + 2x strength if both claws hit (and no TWF penalties), which is probably better than you would get with a greatsword. You don't really get a downside until you get the third iterative, and even then, +0/+0 will often hit more frequently than +0/-5/-10.
I would probably change the duration to something like 1+cha + 2*level rounds per day. Even with a charisma of 16, that is six rounds at first level, which is more than the number of non-cantrips the sorcerer can cast. By fifth level, we are at 14 rounds a day, which would cover most of the combat. (A barbarian with a similar Con of 16 could only rage for 19 rounds.)