The only answer is time, give yourself time to readjust your technique. I would suggest you start playing closer to your original method (at least the soft touch side of things), do not aim to acquire the even sound you produce from your older keyboard or force your fingers to press down the notes this will deplete your endurance too fast and your hand does not learn much when it is stressed out.
Play normally and if your touch is too light on the weighted keys that is ok, if some notes sound too soft or barely played this is fine, the more you practice in this realm the small changes you will make eventually as your fingers become more accustomed to the heaver action will improve your ability to deal with a heavier action. Focus playing parts which cause most tension in your hand and question why it is sapping your energy. The technique compass for you is your old keyboard you practiced on, the effortlessness and ease of playing you felt from that needs to be transfered to the weighted keys.
Playing on an unweighted piano allow us to play with technical inefficiencies which are not punished. Isolated finger movements for instance are not punished enough on unweighted keys, if you catch yourself doing too much of this on a real piano you start tiring out quite fast so keep an eye out for this. When you start to feel tired you have to stop and recover, once the tired burn disappears then start again. When your body starts to tired out and get that burning sensation this is a natural defense mechanism for the body. Our body is preventing us damage during extreme exertion by slowing the key systems needed to maintain muscle contraction, with improvements to your technique you will be able to relax the muscles more and more. We should certainly not push too far with the hands/arms. Sometimes practice with one hand while you rest the other. Push through that burning now and then since you have to develop your physical structure to play a real piano since you are existed in a physically stress free zone for so long.
On a real piano inefficiencies are punished by tiring the hand, of course no matter how good your technique is you will always have some form of energy sapping while playing but we can stay within a certain zone which still allows us to maintain relaxation and flexibility. When you are tired out in a bad zone you feel unrelaxed and inflexible (fingering, notes and expression cannot bend to your will and merely struggle to produce the notes themselves), you tend to struggle to produce your desired sound because the physical aspect of playing is becoming more and more to the front of our attention.
I personally sometimes practice on one digital piano at my home which is touch sensitive but extremely light to play. This allows me to practice fingering and notes and have almost zero effort to produce it since the notes are so easy to sound. So a keyboard with extremely light action can be useful when you want to practice simply notes and fingering (but this is not advised for developing pianists since learning how to play a real piano while practicing is a very important part of their learning.)
Topic: "Funny" problem - switching from an old keyboard