Question: It seems as if both words mean *to humiliate and degrade*?

Answer: 
To *abase* is to behave in a way so as to belittle or degrade someone.

To *abash* is to cause someone to feel embarrassed or ashamed.
The main difference is that 'abase' is something you do yourself (or other people do themselves, of their own volition), whereas 'abash' is something that you do to other people (or other people do to you).

I'm also not convinced that 'abase' really means 'humiliate' or 'degrade', unless you assume that any signs of abasement are automatically humiliating and degrading. That was certainly not the original meaning of 'abase'; I'm not sure it is yet an acquired meaning.

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From an ancient Chambers Twentieth Century Dictionary (1978 reprint of 1972 edition with 1977 supplement), page 1:

**abase**, *v.t.* to lower: to cast down: to humble: to degrade. -- *adj.* **abased**, lowered. -- *n.* **abasement**. [O.Fr. *abaisser*, to bring low--L. *ad*, to, L.L. *bassus*, low.]

**abash**, *v.t.* to strike with shame; to put out of countenance: to astound: to confound.-- *adjs*. **abashed**: **abashless**, shameless: unabashed.-- *n.* **abashment**. [O.Fr. *esbahir* -- pfx. *es-* (L. *ex*, out), *bahir*, to astound--interj. *bah*.]

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It seems clear to me that you do not normally abash yourself, so someone else does it to you.

It is not so clear that you do normally abase yourself, but the verb is often used reflexively - 'he abased himself'. I'm not sure how you could abase someone else.