I suspect that the problem is that your running economy is poor relative to your hiking economy. This means it costs you a lot of energy (and O2) to run and so your hr is higher, maybe even for the same speeds you could hike.
Running is less economical than walking for most people up to about 6km/hr. Then running becomes more economical. Everyone will be a bit different of course and this is a generalization. you can experiment on yourself but walking at this snail like pace and see if you HR drops compared to running at the same slow pace.
With all those km in the past few months I don’t see how you could have ADS.
Did you do the lactate test hiking or running? You need to do the test using the modality you want to train.
To improve running economy you need to run and it can be difficult in a situation like yours. What we do in these situations is to use a run/walk progression. You might start with 1-2 min run with 3-4 min walk repeated for as long as you feel comfortable. over the course of weeks you can increase the run time and decrease the walk time until you are running. During the run times do not work about HR.