If you see only calendar at the page, so then probably you have inserted the shortcode for only "availability calendar"
[bookingcalendar type=1 nummonths=2]
instead of shortcode for booking form
[booking type=1 nummonths=2]
Please open the page with booking form for editing, then remove exist shortcode. After this switch from the "Visual" view mode to "Text" view mode your content editor. And then reinsert the Booking Calendar shortcode and test it.
You need to insert this shortcode:
Please note, this shortcode [bookingcalendar type=1 nummonths=2] is only for showing availability calendar, without form, where you can not select the dates.