MMSI numbers are assigned to the boat, not the radio. So if you bought a used boat and the radio came with a MMSI number programmed into it, then that number should still be good.
The problem is that you will not be able to update the info. So applying for a new number is OK.
However, if you have the Eclipse+ GX1250SA you do not have a DSC radio. Your radio was type accepted before 1999. The manual for it is here.
Do you see a red plastic cover that says "distress" on it? The cover should flip up and under it will be a button. If you do not see that then you do not have a DSC radio. Therefore you will not be able to program a MMSI number into it.
If this is the radio you have, you need to buy a new one. Off the top of my head I would suggest the Quest X. It is the same size as your radio, so it will fit in the same bracket or cutout if you flush mounted. The Quest (as opposed to the Quest X) and the GX1000S are also good. For that matter, any DSC radio in this price range is good.
I did a quick check and Turtle Marine is closing out the Quest radios for $70. Not a bad price. The Quest X can be found for around $120. I like the Quest X over the Quest because the X model is a class D radio, which means it is ALWAYS listening to channel 70. The Quest scans channel 70. But both radios are good and both will work fine.
You will also need a GPS chart plotter on your boat. The radio connects to the plotter through a very simple connection.