Προσπαθώ να μην ξοδεύω άσκοπα, γιατί θέλω να εξοικονομώ λίγα χρήματα κάθε μήνα για το ταξίδι μου.

Because προσπαθώ is normally followed by να + verb:

• Προσπαθώ να διαβάζω. = I try to read.
• Προσπαθώ να μην αργώ. = I try not to be late.

When you negate a να-clause, Greek uses μη(ν), not δεν.

So:

• προσπαθώ να ξοδεύω = I try to spend
• προσπαθώ να μην ξοδεύω = I try not to spend

This is one of the most important patterns in Greek:

• δεν negates ordinary finite verbs
• μη(ν) negates να clauses, imperatives, and some other non-indicative forms

So προσπαθώ να δεν ξοδεύω would be wrong.

Here the present tense shows habitual or repeated action.

The sentence is about a general habit:

• not spending carelessly
• saving some money every month

So ξοδεύω and εξοικονομώ mean something like:

• to be spending / to spend regularly
• to be saving / to save regularly

Because the sentence includes κάθε μήνα (every month), the idea is clearly repeated over time, not a one-time action.

If you used a more one-off form such as an aorist subjunctive, it would sound more like a single completed act rather than an ongoing habit.