To notate melodies with scale degrees there is a number based notation that I know of (there may be others), coming from the Hungarian/Kodaly school of musical pedagogy. Actually, originally the so called "moving do" system is used, but in countries where note names are latin based (do, re, mi...), moving do notation becomes very confusing and so numbers are used instead. Either with moving do or with numbers, upper and lower apostrophes are used to denote octaves. For example, this melody in D major
would be notated as (comma used in lieu of lower apostrophe, to denote lower octave)
With moving do - D: sol mi sol do' do ti, la, sol,
with numbers - D: 5 3 5 1' 1 7, 6, 5,
NOTE: as you can see from the example, 1 (or "do") is NOT the first note of the melody, it is the 1st degree of the scale of the key you're working with.
This notation is also quite useful to quickly notate a short melody when there's no staff paper at hand (or when writing text in a word processor). Rhythm, if required, can be notated above the note numbers.
Indeed, melodic dictation by scale degrees is a very useful hearing training exercise, and, among other benefits, can be a stepping stone towards melodic dictation on the staff, which is the more usual (or, at least, traditional) way of doing melodic dictation.
If writing on the staff is the end result intended (typically it is, in exam situations), then writing down the scale degrees rather than just putting down the notes directly on the staff, is an unnecessary intermediate step. You can do it initially as an exercise, but it will be very difficult to acquire the necessary speed if you keep doing it like that.
If it currently is too difficult to rapidly translate the knowledge of the heard note into the respective position on the staff, then you absolutely must practice that before, or in parallel, with the auditive practice per se. Practice solfegio (saying out loud the note names of written melodies) and write scales and arpeggios saying the note names as you write them on the staff. You'll quickly be able to, when identifying a note as C, notate a C on the staff rather than writing "1" to later translate it to the right position on the staff. Or, when hearing a sequence of joint scale degrees, quickly jotting them down on the staff, etc.
You can use light dots or dashes in pencil to notate the first "anchor points", as you rightly call them, or your first impression of a note, and on the following hearing confirm or correct your impression and refine the notation with the proper note shape, stems, etc.
Now, if your difficulty relies on associating scale degrees with note names in the different keys (e.g. D - E - F# etc. in D major, rather than 1, 2, 3), then, again you must work on your knowledge of the 12 keys and respective key signatures (if you're only required to know a few keys or even just the C key for now, than this step can be done later on).
So, your steps 1 to 3 are fine, but you have to work towards doing them directly on the staff. With experience you'll find that in step 1 you'll be able to write already some note information (where the tonic appears, etc.). And, with more complex rhythms you may not be able to capture all rhythm information in the first hearing, and that's normal.
