A few words of advice about taking the pistol's apart.
Although the screws look like they are the same, some of them are slightly different in length so it's a good idea to screw them back into the holes they came out of after the part they were holding are removed.
Start with the two screws on either side of the hammer and then the one screw on the bottom of the grip. With these removed, the grip strap and grip will pull off.
Loosen but don't remove the screw that holds the mainspring in place. Then, rotate the mainspring sideways slightly to disengage it from the hammer.
Remove the screws on the bottom of the trigger guard and take it off.
Under the trigger guard you will see a flat, two bladed spring. Pay special attention to the way it is resting and how the two leaves of the spring are located. The spring is not symmetrical and if it is put back in place upside down the trigger and cylinder lock won't work.
Remove the trigger screw and the trigger. This will also allow the cylinder lock to be removed.
Remove the big screw that the hammer pivots on. Carefully slide the hammer downward and out of the frame. Note: the cylinder hand that rotates the cylinder is attached to the hammer and it will pull out of the frame along with the hammer. There should be a small spring on the back of the cylinder hand. This little spring will often break and if it does break the cylinder hand won't reliably rotate the cylinder. It has nothing to do with the hammer staying fully cocked.
Clean off any old, dried grease or oil from the parts, re-oil with a light weight gun oil and things should be ready to go back together.
While the parts are out of the gun look at the notches on the front of the hammer. These should be sharp without any chips in them.
Look at the spur on the upper end of the trigger. This should also be sharp, not rounded off. It is what engages the notches in the hammer to hold the gun at half and full cock.
If everything looks good, reassemble the gun and see if cleaning it up fixed the problem.
Spare parts for your gun can be bought from Dixie Gun Works. They might need a bit of "fitting" but they should do the job if they are needed.