Looks like either the springs are too loose or weak, or the six screws on top are too tight. If you can't push it down with your hand it's the latter.
Measure... To avoid redrilling the holes you need to know the diameter and length of the string capstans so they'll fit into the original capstan bushings - or you'll also need to know the outer diameter and length of the bushings so they'll fit the existing holes. And then you need to know the size of the tuner housing so they'll match the spacing. Your photos don't show how the tuners are attached but I assume there's a screw or two. You may be able to find a matching set but if the only issue is misplaced screw holes, filling them and drilling new ones is no big issue. Gotoh, Kluson and Wilkinson among others make decent priced good quality sets, I'd say for non-locking tuners the price range for a decent quality set is about $30-40, premium quality being double or triple of that and most likely overkill for that guitar.
The type of tuners you need is 6L meaning 6 in line on the left side looking from the top.