Hello dl49570,
I´ve just tested the approach 35L with the B737-800. I´ve selected on the map the ILS frequency and switched the Capt EFIS to APP mode. Firstly I used the VOR-LOC button to be sure that You are at the centerline. Secondly, short before the ILS initialpoint You can Switch to APP mode. Then the ILS aproach is full established and the touch down will be done properly.
Several times I´ve observed the same reaction what You descripted. It may be this is a bug in X-Plane 11. The work around is as followes. Make a go around and shortly before You reached the ILS initial point, Switch off the Autopilot and then again, switch on firstly flight manager, secondly speed, thirdly VOR-LOC or direct APP. After this, don´t forget the CMD button. Allways have a look that You adjust the correct ILS frequency for RWY 35L using the map.
Have a nice flight,
All best regards,Horst K