New GFCI keeps tripping
I notice you have 4 wires on this GFCI using all 4 terminals.  

Two of them, the LINE terminals, are connected to the power source, and will allow this GFCI to power up and do its thing.  Hook up only these wires and tape off the others.  Does the GFCI power up and behave correctly? 

The other two, LOAD, are not there for convenience.  They have a very specific job. They extend the GFCI's "zone of protection" to other loads.  Doing this is both a blessing and a curse: it confers GFCI protection to the loads; but it means this GFCI will trip if the downline load has a ground fault.  That is why these terminals should not be used lightly or frivolously. 

The reason it trips now, when it didn't before, is that it didn't have a ground fault before, and now it does.  The old GFCI's test button wasn't stuck, it was refusing to reset because of the ground fault.  That "retest  before allowing reset" is a more advanced feature of better GFCIs.  

If you followed my instructions above, some things in your house are now out.  One of them is the culprit: the source of the ground fault.  Once it is removed from the circuit, the GFCI will reset.   

So you can reconnect the LOAD wires and start unhooking each of the possible culprits until you find the one responsible.